GCSE Maths: Solving linear inequalities (Higher)

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x + 5 > 9
2. Get the plain numbers on the other side. Subtract 5 from both sides. This is the inverse of adding 5.
   x > 4
3. State the solution. This is the complete set of values that make the inequality true.
   x > 4

Check. Test a value inside the solution: x = 5 gives left side = 5 + 5 = 10, right side = 9. Is 10 > 9? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 3 + 5 = 8, right side = 9. Is 8 > 9? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x − 3 ≤ 4
2. Get the plain numbers on the other side. Add 3 to both sides. This is the inverse of subtracting 3.
   x ≤ 7
3. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 7

Check. Test a value inside the solution: x = 6 gives left side = 6 − 3 = 3, right side = 4. Is 3 ≤ 4? Yes, it is true. ✓
Test a value outside it: x = 8 gives left side = 8 − 3 = 5, right side = 4. Is 5 ≤ 4? No, it is false, so 8 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x < 14
2. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x < 14 ÷ 2 = 7
3. State the solution. This is the complete set of values that make the inequality true.
   x < 7

Check. Test a value inside the solution: x = 6 gives left side = 2 × 6 = 12, right side = 14. Is 12 < 14? Yes, it is true. ✓
Test a value outside it: x = 8 gives left side = 2 × 8 = 16, right side = 14. Is 16 < 14? No, it is false, so 8 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x4 ≥ 3
2. Multiply both sides by 4 to get rid of the fraction. Multiplying by the denominator cancels the division. 4 is a positive number, so the inequality sign does not change.
   x ≥ 12
3. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 12

Check. Test a value inside the solution: x = 13 gives left side = 134 = 3.25, right side = 3. Is 3.25 ≥ 3? Yes, it is true. ✓
Test a value outside it: x = 11 gives left side = 114 = 2.75, right side = 3. Is 2.75 ≥ 3? No, it is false, so 11 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x + 7 ≤ 2
2. Get the plain numbers on the other side. Subtract 7 from both sides. This is the inverse of adding 7.
   x ≤ −5
3. State the solution. This is the complete set of values that make the inequality true.
   x ≤ −5

Check. Test a value inside the solution: x = −6 gives left side = (−6) + 7 = 1, right side = 2. Is 1 ≤ 2? Yes, it is true. ✓
Test a value outside it: x = −4 gives left side = (−4) + 7 = 3, right side = 2. Is 3 ≤ 2? No, it is false, so −4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   5x > 35
2. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x > 35 ÷ 5 = 7
3. State the solution. This is the complete set of values that make the inequality true.
   x > 7

Check. Test a value inside the solution: x = 8 gives left side = 5 × 8 = 40, right side = 35. Is 40 > 35? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 5 × 6 = 30, right side = 35. Is 30 > 35? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x + 3 > 11
2. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   2x > 8
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x > 8 ÷ 2 = 4
4. State the solution. This is the complete set of values that make the inequality true.
   x > 4

Check. Test a value inside the solution: x = 5 gives left side = 2 × 5 + 3 = 13, right side = 11. Is 13 > 11? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 2 × 3 + 3 = 9, right side = 11. Is 9 > 11? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   3x − 4 ≤ 14
2. Get the plain numbers on the other side. Add 4 to both sides. This is the inverse of subtracting 4.
   3x ≤ 18
3. Divide both sides by 3. 3x means 3 lots of x. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   x ≤ 18 ÷ 3 = 6
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 6

Check. Test a value inside the solution: x = 5 gives left side = 3 × 5 − 4 = 11, right side = 14. Is 11 ≤ 14? Yes, it is true. ✓
Test a value outside it: x = 7 gives left side = 3 × 7 − 4 = 17, right side = 14. Is 17 ≤ 14? No, it is false, so 7 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   5x + 1 < 26
2. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   5x < 25
3. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x < 25 ÷ 5 = 5
4. State the solution. This is the complete set of values that make the inequality true.
   x < 5

Check. Test a value inside the solution: x = 4 gives left side = 5 × 4 + 1 = 21, right side = 26. Is 21 < 26? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 5 × 6 + 1 = 31, right side = 26. Is 31 < 26? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   4x − 7 ≥ 9
2. Get the plain numbers on the other side. Add 7 to both sides. This is the inverse of subtracting 7.
   4x ≥ 16
3. Divide both sides by 4. 4x means 4 lots of x. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.
   x ≥ 16 ÷ 4 = 4
4. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 4

Check. Test a value inside the solution: x = 5 gives left side = 4 × 5 − 7 = 13, right side = 9. Is 13 ≥ 9? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 4 × 3 − 7 = 5, right side = 9. Is 5 ≥ 9? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x + 9 ≤ 3
2. Get the plain numbers on the other side. Subtract 9 from both sides. This is the inverse of adding 9.
   2x ≤ −6
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x ≤ −6 ÷ 2 = −3
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ −3

Check. Test a value inside the solution: x = −4 gives left side = 2 × (−4) + 9 = 1, right side = 3. Is 1 ≤ 3? Yes, it is true. ✓
Test a value outside it: x = −2 gives left side = 2 × (−2) + 9 = 5, right side = 3. Is 5 ≤ 3? No, it is false, so −2 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   6x − 5 > −17
2. Get the plain numbers on the other side. Add 5 to both sides. This is the inverse of subtracting 5.
   6x > −12
3. Divide both sides by 6. 6x means 6 lots of x. The inverse of multiplying by 6 is dividing by 6. 6 is positive, so the inequality sign stays the same.
   x > −12 ÷ 6 = −2
4. State the solution. This is the complete set of values that make the inequality true.
   x > −2

Check. Test a value inside the solution: x = −1 gives left side = 6 × (−1) − 5 = −11, right side = −17. Is −11 > −17? Yes, it is true. ✓
Test a value outside it: x = −3 gives left side = 6 × (−3) − 5 = −23, right side = −17. Is −23 > −17? No, it is false, so −3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   12 > 2x + 4
2. Get the plain numbers on the other side. Subtract 4 from both sides.
   8 > 2x
3. Divide both sides by 2. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   8 ÷ 2 = 4
   4 > x
4. Write it with x on the left. 4 > x says the same thing as x < 4: the sign still points to the smaller number.
   x < 4
5. State the solution. This is the complete set of values that make the inequality true.
   x < 4

Check. Test a value inside the solution: x = 3 gives left side = 12, right side = 2 × 3 + 4 = 10. Is 12 > 10? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 12, right side = 2 × 5 + 4 = 14. Is 12 > 14? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   20 ≤ 3x − 1
2. Get the plain numbers on the other side. Add 1 to both sides.
   21 ≤ 3x
3. Divide both sides by 3. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   21 ÷ 3 = 7
   7 ≤ x
4. Write it with x on the left. 7 ≤ x says the same thing as x ≥ 7: the sign still points to the smaller number.
   x ≥ 7
5. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 7

Check. Test a value inside the solution: x = 8 gives left side = 20, right side = 3 × 8 − 1 = 23. Is 20 ≤ 23? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 20, right side = 3 × 6 − 1 = 17. Is 20 ≤ 17? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   7 < 5x − 8
2. Get the plain numbers on the other side. Add 8 to both sides.
   15 < 5x
3. Divide both sides by 5. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   15 ÷ 5 = 3
   3 < x
4. Write it with x on the left. 3 < x says the same thing as x > 3: the sign still points to the smaller number.
   x > 3
5. State the solution. This is the complete set of values that make the inequality true.
   x > 3

Check. Test a value inside the solution: x = 4 gives left side = 7, right side = 5 × 4 − 8 = 12. Is 7 < 12? Yes, it is true. ✓
Test a value outside it: x = 2 gives left side = 7, right side = 5 × 2 − 8 = 2. Is 7 < 2? No, it is false, so 2 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   10 ≥ 4x + 2
2. Get the plain numbers on the other side. Subtract 2 from both sides.
   8 ≥ 4x
3. Divide both sides by 4. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.
   8 ÷ 4 = 2
   2 ≥ x
4. Write it with x on the left. 2 ≥ x says the same thing as x ≤ 2: the sign still points to the smaller number.
   x ≤ 2
5. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 2

Check. Test a value inside the solution: x = 1 gives left side = 10, right side = 4 × 1 + 2 = 6. Is 10 ≥ 6? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 10, right side = 4 × 3 + 2 = 14. Is 10 ≥ 14? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   −4 < 2x + 6
2. Get the plain numbers on the other side. Subtract 6 from both sides.
   −10 < 2x
3. Divide both sides by 2. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   −10 ÷ 2 = −5
   −5 < x
4. Write it with x on the left. −5 < x says the same thing as x > −5: the sign still points to the smaller number.
   x > −5
5. State the solution. This is the complete set of values that make the inequality true.
   x > −5

Check. Test a value inside the solution: x = −4 gives left side = −4, right side = 2 × (−4) + 6 = −2. Is −4 < −2? Yes, it is true. ✓
Test a value outside it: x = −6 gives left side = −4, right side = 2 × (−6) + 6 = −6. Is −4 < −6? No, it is false, so −6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2 > 3x − 7
2. Get the plain numbers on the other side. Add 7 to both sides.
   9 > 3x
3. Divide both sides by 3. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   9 ÷ 3 = 3
   3 > x
4. Write it with x on the left. 3 > x says the same thing as x < 3: the sign still points to the smaller number.
   x < 3
5. State the solution. This is the complete set of values that make the inequality true.
   x < 3

Check. Test a value inside the solution: x = 2 gives left side = 2, right side = 3 × 2 − 7 = −1. Is 2 > −1? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 2, right side = 3 × 4 − 7 = 5. Is 2 > 5? No, it is false, so 4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x3 − 2 > 1
2. Multiply both sides by 3 to get rid of the fraction. Multiplying by the denominator cancels the division. 3 is a positive number, so the inequality sign does not change.
   x − 6 > 3
3. Get the plain numbers on the other side. Add 6 to both sides. This is the inverse of subtracting 6.
   x > 9
4. State the solution. This is the complete set of values that make the inequality true.
   x > 9

Check. Test a value inside the solution: x = 10 gives left side = 103 − 2 = 43, right side = 1. Is 43 > 1? Yes, it is true. ✓
Test a value outside it: x = 8 gives left side = 83 − 2 = 23, right side = 1. Is 23 > 1? No, it is false, so 8 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x2 + 3 ≤ 5
2. Multiply both sides by 2 to get rid of the fraction. Multiplying by the denominator cancels the division. 2 is a positive number, so the inequality sign does not change.
   x + 6 ≤ 10
3. Get the plain numbers on the other side. Subtract 6 from both sides. This is the inverse of adding 6.
   x ≤ 4
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 4

Check. Test a value inside the solution: x = 3 gives left side = 32 + 3 = 4.5, right side = 5. Is 4.5 ≤ 5? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 52 + 3 = 5.5, right side = 5. Is 5.5 ≤ 5? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x5 + 1 ≥ 4
2. Multiply both sides by 5 to get rid of the fraction. Multiplying by the denominator cancels the division. 5 is a positive number, so the inequality sign does not change.
   x + 5 ≥ 20
3. Get the plain numbers on the other side. Subtract 5 from both sides. This is the inverse of adding 5.
   x ≥ 15
4. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 15

Check. Test a value inside the solution: x = 16 gives left side = 165 + 1 = 4.2, right side = 4. Is 4.2 ≥ 4? Yes, it is true. ✓
Test a value outside it: x = 14 gives left side = 145 + 1 = 3.8, right side = 4. Is 3.8 ≥ 4? No, it is false, so 14 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x3 < 8
2. Multiply both sides by 3 to get rid of the fraction. Multiplying by the denominator cancels the division. 3 is a positive number, so the inequality sign does not change.
   2x < 24
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x < 24 ÷ 2 = 12
4. State the solution. This is the complete set of values that make the inequality true.
   x < 12

Check. Test a value inside the solution: x = 11 gives left side = 2 × 113 = 223, right side = 8. Is 223 < 8? Yes, it is true. ✓
Test a value outside it: x = 13 gives left side = 2 × 133 = 263, right side = 8. Is 263 < 8? No, it is false, so 13 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x + 1 > 9
2. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   2x > 8
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x > 8 ÷ 2 = 4
4. State the solution. This is the complete set of values that make the inequality true.
   x > 4
   On a number line: an open circle at 4 with an arrow pointing right.

Check. Test a value inside the solution: x = 5 gives left side = 2 × 5 + 1 = 11, right side = 9. Is 11 > 9? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 2 × 3 + 1 = 7, right side = 9. Is 7 > 9? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   3x − 2 ≤ 7
2. Get the plain numbers on the other side. Add 2 to both sides. This is the inverse of subtracting 2.
   3x ≤ 9
3. Divide both sides by 3. 3x means 3 lots of x. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   x ≤ 9 ÷ 3 = 3
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 3
   On a number line: a closed (filled) circle at 3 with an arrow pointing left.

Check. Test a value inside the solution: x = 2 gives left side = 3 × 2 − 2 = 4, right side = 7. Is 4 ≤ 7? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 3 × 4 − 2 = 10, right side = 7. Is 10 ≤ 7? No, it is false, so 4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   4x + 5 < 1
2. Get the plain numbers on the other side. Subtract 5 from both sides. This is the inverse of adding 5.
   4x < −4
3. Divide both sides by 4. 4x means 4 lots of x. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.
   x < −4 ÷ 4 = −1
4. State the solution. This is the complete set of values that make the inequality true.
   x < −1
   On a number line: an open circle at −1 with an arrow pointing left.

Check. Test a value inside the solution: x = −2 gives left side = 4 × (−2) + 5 = −3, right side = 1. Is −3 < 1? Yes, it is true. ✓
Test a value outside it: x = 0 gives left side = 4 × 0 + 5 = 5, right side = 1. Is 5 < 1? No, it is false, so 0 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   5x − 1 ≥ 14
2. Get the plain numbers on the other side. Add 1 to both sides. This is the inverse of subtracting 1.
   5x ≥ 15
3. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x ≥ 15 ÷ 5 = 3
4. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 3
   On a number line: a closed (filled) circle at 3 with an arrow pointing right.

Check. Test a value inside the solution: x = 4 gives left side = 5 × 4 − 1 = 19, right side = 14. Is 19 ≥ 14? Yes, it is true. ✓
Test a value outside it: x = 2 gives left side = 5 × 2 − 1 = 9, right side = 14. Is 9 ≥ 14? No, it is false, so 2 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x + 3 < 12
2. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   2x < 9
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x < 9 ÷ 2 = 4.5
4. State the solution. This is the complete set of values that make the inequality true.
   x < 4.5
5. Find the largest whole number that fits. The solution says x is less than 4.5. A whole number must be below 4.5. 4.5 is not a whole number, so round down.
   4

Check. Test a value inside the solution: x = 4 gives left side = 2 × 4 + 3 = 11, right side = 12. Is 11 < 12? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 2 × 5 + 3 = 13, right side = 12. Is 13 < 12? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   5x − 1 ≤ 24
2. Get the plain numbers on the other side. Add 1 to both sides. This is the inverse of subtracting 1.
   5x ≤ 25
3. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x ≤ 25 ÷ 5 = 5
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 5
5. Find the largest whole number that fits. The solution says x is less than or equal to 5. A whole number must be below 5, or equal to it. 5 itself is allowed.
   5

Check. Test a value inside the solution: x = 4 gives left side = 5 × 4 − 1 = 19, right side = 24. Is 19 ≤ 24? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 5 × 6 − 1 = 29, right side = 24. Is 29 ≤ 24? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   3x + 2 < 9
2. Get the plain numbers on the other side. Subtract 2 from both sides. This is the inverse of adding 2.
   3x < 7
3. Divide both sides by 3. 3x means 3 lots of x. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   x < 7 ÷ 3 = 73
4. State the solution. This is the complete set of values that make the inequality true.
   x < 73
5. Find the largest whole number that fits. The solution says x is less than 73. A whole number must be below 73. 73 is not a whole number, so round down.
   2

Check. Test a value inside the solution: x = 2 gives left side = 3 × 2 + 2 = 8, right side = 9. Is 8 < 9? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 3 × 3 + 2 = 11, right side = 9. Is 11 < 9? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   4x + 1 ≤ 20
2. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   4x ≤ 19
3. Divide both sides by 4. 4x means 4 lots of x. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.
   x ≤ 19 ÷ 4 = 4.75
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 4.75
5. Find the largest whole number that fits. The solution says x is less than or equal to 4.75. A whole number must be below 4.75, or equal to it. 4.75 is not a whole number, so round down.
   4

Check. Test a value inside the solution: x = 4 gives left side = 4 × 4 + 1 = 17, right side = 20. Is 17 ≤ 20? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 4 × 5 + 1 = 21, right side = 20. Is 21 ≤ 20? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   3x + 1 > 13
2. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   3x > 12
3. Divide both sides by 3. 3x means 3 lots of x. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   x > 12 ÷ 3 = 4
4. State the solution. This is the complete set of values that make the inequality true.
   x > 4
5. Find the smallest whole number that fits. The solution says x is greater than 4. A whole number must be above 4. 4 itself is not allowed.
   5

Check. Test a value inside the solution: x = 5 gives left side = 3 × 5 + 1 = 16, right side = 13. Is 16 > 13? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 3 × 3 + 1 = 10, right side = 13. Is 10 > 13? No, it is false, so 3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x − 5 ≥ 4
2. Get the plain numbers on the other side. Add 5 to both sides. This is the inverse of subtracting 5.
   2x ≥ 9
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x ≥ 9 ÷ 2 = 4.5
4. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 4.5
5. Find the smallest whole number that fits. The solution says x is greater than or equal to 4.5. A whole number must be above 4.5, or equal to it. 4.5 is not a whole number, so round up.
   5

Check. Test a value inside the solution: x = 5 gives left side = 2 × 5 − 5 = 5, right side = 4. Is 5 ≥ 4? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 2 × 4 − 5 = 3, right side = 4. Is 3 ≥ 4? No, it is false, so 4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   5x + 2 > 6
2. Get the plain numbers on the other side. Subtract 2 from both sides. This is the inverse of adding 2.
   5x > 4
3. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x > 4 ÷ 5 = 0.8
4. State the solution. This is the complete set of values that make the inequality true.
   x > 0.8
5. Find the smallest whole number that fits. The solution says x is greater than 0.8. A whole number must be above 0.8. 0.8 is not a whole number, so round up.
   1

Check. Test a value inside the solution: x = 1 gives left side = 5 × 1 + 2 = 7, right side = 6. Is 7 > 6? Yes, it is true. ✓
Test a value outside it: x = 0 gives left side = 5 × 0 + 2 = 2, right side = 6. Is 2 > 6? No, it is false, so 0 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x + 1 < 11
2. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   2x < 10
3. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x < 10 ÷ 2 = 5
4. State the solution. This is the complete set of values that make the inequality true.
   x < 5
5. List the positive whole numbers that fit. Positive means 1, 2, 3, ... . Go up in ones until you reach the boundary 5 (which is not allowed).
   1, 2, 3, 4

Check. Test a value inside the solution: x = 4 gives left side = 2 × 4 + 1 = 9, right side = 11. Is 9 < 11? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 2 × 6 + 1 = 13, right side = 11. Is 13 < 11? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   3x ≤ 12
2. Divide both sides by 3. 3x means 3 lots of x. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
   x ≤ 12 ÷ 3 = 4
3. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 4
4. List the positive whole numbers that fit. Positive means 1, 2, 3, ... . Go up in ones until you reach the boundary 4.
   1, 2, 3, 4

Check. Test a value inside the solution: x = 3 gives left side = 3 × 3 = 9, right side = 12. Is 9 ≤ 12? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 3 × 5 = 15, right side = 12. Is 15 ≤ 12? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   5x − 3 < 12
2. Get the plain numbers on the other side. Add 3 to both sides. This is the inverse of subtracting 3.
   5x < 15
3. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x < 15 ÷ 5 = 3
4. State the solution. This is the complete set of values that make the inequality true.
   x < 3
5. List the positive whole numbers that fit. Positive means 1, 2, 3, ... . Go up in ones until you reach the boundary 3 (which is not allowed).
   1, 2

Check. Test a value inside the solution: x = 2 gives left side = 5 × 2 − 3 = 7, right side = 12. Is 7 < 12? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 5 × 4 − 3 = 17, right side = 12. Is 17 < 12? No, it is false, so 4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   4x + 3 ≤ 20
2. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   4x ≤ 17
3. Divide both sides by 4. 4x means 4 lots of x. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.
   x ≤ 17 ÷ 4 = 4.25
4. State the solution. This is the complete set of values that make the inequality true.
   x ≤ 4.25
5. List the positive whole numbers that fit. Positive means 1, 2, 3, ... . Go up in ones until you reach the boundary 4.25.
   1, 2, 3, 4

Check. Test a value inside the solution: x = 4 gives left side = 4 × 4 + 3 = 19, right side = 20. Is 19 ≤ 20? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 4 × 5 + 3 = 23, right side = 20. Is 23 ≤ 20? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x + 32 ≥ 4
2. Multiply both sides by 2 to get rid of the fraction. Multiplying by the denominator cancels the division. 2 is a positive number, so the inequality sign does not change.
   x + 3 ≥ 8
3. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   x ≥ 5
4. State the solution. This is the complete set of values that make the inequality true.
   x ≥ 5

Check. Test a value inside the solution: x = 6 gives left side = 6 + 32 = 4.5, right side = 4. Is 4.5 ≥ 4? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 4 + 32 = 3.5, right side = 4. Is 3.5 ≥ 4? No, it is false, so 4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x − 13 < 5
2. Multiply both sides by 3 to get rid of the fraction. Multiplying by the denominator cancels the division. 3 is a positive number, so the inequality sign does not change.
   2x − 1 < 15
3. Get the plain numbers on the other side. Add 1 to both sides. This is the inverse of subtracting 1.
   2x < 16
4. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
   x < 16 ÷ 2 = 8
5. State the solution. This is the complete set of values that make the inequality true.
   x < 8

Check. Test a value inside the solution: x = 7 gives left side = 2 × 7 − 13 = 133, right side = 5. Is 133 < 5? Yes, it is true. ✓
Test a value outside it: x = 9 gives left side = 2 × 9 − 13 = 173, right side = 5. Is 173 < 5? No, it is false, so 9 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x + 45 > 3
2. Multiply both sides by 5 to get rid of the fraction. Multiplying by the denominator cancels the division. 5 is a positive number, so the inequality sign does not change.
   x + 4 > 15
3. Get the plain numbers on the other side. Subtract 4 from both sides. This is the inverse of adding 4.
   x > 11
4. State the solution. This is the complete set of values that make the inequality true.
   x > 11

Check. Test a value inside the solution: x = 12 gives left side = 12 + 45 = 3.2, right side = 3. Is 3.2 > 3? Yes, it is true. ✓
Test a value outside it: x = 10 gives left side = 10 + 45 = 2.8, right side = 3. Is 2.8 > 3? No, it is false, so 10 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   x + 32 ≥ x − 13
2. Multiply both sides by 6 to get rid of the fraction. Multiplying by the denominator cancels the division. 6 is a positive number, so the inequality sign does not change.
   3(x + 3) ≥ 2(x − 1)
3. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   3 × x = 3x
   3 × 3 = 9
   2 × x = 2x
   2 × (−1) = −2
   3x + 9 ≥ 2x − 2
4. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
   Subtract 2x from both sides:
   x + 9 ≥ −2
5. Get the plain numbers on the other side. Subtract 9 from both sides. This is the inverse of adding 9.
   x ≥ −11
6. State the solution. This is the complete set of values that make the inequality true.
   x ≥ −11

Check. Test a value inside the solution: x = −10 gives left side = (−10) + 32 = −3.5, right side = (−10) − 13 = −113. Is −3.5 ≥ −113? Yes, it is true. ✓
Test a value outside it: x = −12 gives left side = (−12) + 32 = −4.5, right side = (−12) − 13 = −133. Is −4.5 ≥ −133? No, it is false, so −12 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   2x − 13 < x + 42
2. Multiply both sides by 6 to get rid of the fraction. Multiplying by the denominator cancels the division. 6 is a positive number, so the inequality sign does not change.
   2(2x − 1) < 3(x + 4)
3. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   2 × 2x = 4x
   2 × (−1) = −2
   3 × x = 3x
   3 × 4 = 12
   4x − 2 < 3x + 12
4. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
   Subtract 3x from both sides:
   x − 2 < 12
5. Get the plain numbers on the other side. Add 2 to both sides. This is the inverse of subtracting 2.
   x < 14
6. State the solution. This is the complete set of values that make the inequality true.
   x < 14

Check. Test a value inside the solution: x = 13 gives left side = 2 × 13 − 13 = 253, right side = 13 + 42 = 8.5. Is 253 < 8.5? Yes, it is true. ✓
Test a value outside it: x = 15 gives left side = 2 × 15 − 13 = 293, right side = 15 + 42 = 9.5. Is 293 < 9.5? No, it is false, so 15 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
   3x + 14 > x − 23
2. Multiply both sides by 12 to get rid of the fraction. Multiplying by the denominator cancels the division. 12 is a positive number, so the inequality sign does not change.
   3(3x + 1) > 4(x − 2)
3. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   3 × 3x = 9x
   3 × 1 = 3
   4 × x = 4x
   4 × (−2) = −8
   9x + 3 > 4x − 8
4. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
   Subtract 4x from both sides:
   5x + 3 > −8
5. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   5x > −11
6. Divide both sides by 5. 5x means 5 lots of x. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
   x > −11 ÷ 5 = −2.2
7. State the solution. This is the complete set of values that make the inequality true.
   x > −2.2

Check. Test a value inside the solution: x = −2 gives left side = 3 × (−2) + 14 = −1.25, right side = (−2) − 23 = −43. Is −1.25 > −43? Yes, it is true. ✓
Test a value outside it: x = −3 gives left side = 3 × (−3) + 14 = −2, right side = (−3) − 23 = −53. Is −2 > −53? No, it is false, so −3 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.