GCSE Maths: Negative numbers, brackets and letters on both sides (Foundation)

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 < 8
2. Divide both sides by −2, and reverse the inequality sign. −2x means −2 lots of x, so we divide by −2. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So < becomes >.
   x > 8 ÷ (−2) = −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 = −3 gives left side = −2 × (−3) = 6, right side = 8. Is 6 < 8? Yes, it is true. ✓
Test a value outside it: x = −5 gives left side = −2 × (−5) = 10, right side = 8. Is 10 < 8? No, it is false, so −5 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −2x < 8 this gives −8 < 2x. Divide by 2 (positive, so no flip): −4 < x, which is the same as x > −4. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

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 ≥ 20
2. Divide both sides by −5, and reverse the inequality sign. −5x means −5 lots of x, so we divide by −5. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So ≥ becomes ≤.
   x ≤ 20 ÷ (−5) = −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) = 25, right side = 20. Is 25 ≥ 20? Yes, it is true. ✓
Test a value outside it: x = −3 gives left side = −5 × (−3) = 15, right side = 20. Is 15 ≥ 20? No, it is false, so −3 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −5x ≥ 20 this gives −20 ≥ 5x. Divide by 5 (positive, so no flip): −4 ≥ x, which is the same as x ≤ −4. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

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 > 1
2. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   −x > −2
3. Divide both sides by −1, and reverse the inequality sign. −x means −1 lots of x, so we divide by −1. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So > becomes <.
   x < −2 ÷ (−1) = 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 = −1 + 3 = 2, right side = 1. Is 2 > 1? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = −3 + 3 = 0, right side = 1. Is 0 > 1? No, it is false, so 3 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −x + 3 > 1 this gives 2 > x. Divide by 1 (positive, so no flip): 2 > x, which is the same as x < 2. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

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 + 7 ≤ 1
2. Get the plain numbers on the other side. Subtract 7 from both sides. This is the inverse of adding 7.
   −2x ≤ −6
3. Divide both sides by −2, and reverse the inequality sign. −2x means −2 lots of x, so we divide by −2. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So ≤ becomes ≥.
   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 + 7 = −1, right side = 1. Is −1 ≤ 1? Yes, it is true. ✓
Test a value outside it: x = 2 gives left side = −2 × 2 + 7 = 3, right side = 1. Is 3 ≤ 1? No, it is false, so 2 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −2x + 7 ≤ 1 this gives 6 ≤ 2x. Divide by 2 (positive, so no flip): 3 ≤ x, which is the same as x ≥ 3. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

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 + 10 > 4
2. Get the plain numbers on the other side. Subtract 10 from both sides. This is the inverse of adding 10.
   −3x > −6
3. Divide both sides by −3, and reverse the inequality sign. −3x means −3 lots of x, so we divide by −3. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So > becomes <.
   x < −6 ÷ (−3) = 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 = −3 × 1 + 10 = 7, right side = 4. Is 7 > 4? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = −3 × 3 + 10 = 1, right side = 4. Is 1 > 4? No, it is false, so 3 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −3x + 10 > 4 this gives 6 > 3x. Divide by 3 (positive, so no flip): 2 > x, which is the same as x < 2. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

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 ≥ 13
2. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   −4x ≥ 12
3. Divide both sides by −4, and reverse the inequality sign. −4x means −4 lots of x, so we divide by −4. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So ≥ becomes ≤.
   x ≤ 12 ÷ (−4) = −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 = −4 × (−4) + 1 = 17, right side = 13. Is 17 ≥ 13? Yes, it is true. ✓
Test a value outside it: x = −2 gives left side = −4 × (−2) + 1 = 9, right side = 13. Is 9 ≥ 13? No, it is false, so −2 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −4x + 1 ≥ 13 this gives −12 ≥ 4x. Divide by 4 (positive, so no flip): −3 ≥ x, which is the same as x ≤ −3. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

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 < 3x + 10
2. 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:
   2x + 2 < 10
3. Get the plain numbers on the other side. Subtract 2 from both sides. This is the inverse of adding 2.
   2x < 8
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 < 8 ÷ 2 = 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 = 5 × 3 + 2 = 17, right side = 3 × 3 + 10 = 19. Is 17 < 19? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 5 × 5 + 2 = 27, right side = 3 × 5 + 10 = 25. Is 27 < 25? 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.
   4x − 1 ≥ 2x + 9
2. 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:
   2x − 1 ≥ 9
3. Get the plain numbers on the other side. Add 1 to both sides. This is the inverse of subtracting 1.
   2x ≥ 10
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 ≥ 10 ÷ 2 = 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 = 6 gives left side = 4 × 6 − 1 = 23, right side = 2 × 6 + 9 = 21. Is 23 ≥ 21? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 4 × 4 − 1 = 15, right side = 2 × 4 + 9 = 17. Is 15 ≥ 17? 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.
   7x + 3 > 2x + 13
2. 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:
   5x + 3 > 13
3. Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.
   5x > 10
4. 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 > 10 ÷ 5 = 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 = 3 gives left side = 7 × 3 + 3 = 24, right side = 2 × 3 + 13 = 19. Is 24 > 19? Yes, it is true. ✓
Test a value outside it: x = 1 gives left side = 7 × 1 + 3 = 10, right side = 2 × 1 + 13 = 15. Is 10 > 15? No, it is false, so 1 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 + 5 ≤ 5x + 1
2. 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:
   5 ≤ 2x + 1
3. Get the plain numbers on the other side. Subtract 1 from both sides.
   4 ≤ 2x
4. 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.
   4 ÷ 2 = 2
   2 ≤ x
5. 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
6. 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 = 3 gives left side = 3 × 3 + 5 = 14, right side = 5 × 3 + 1 = 16. Is 14 ≤ 16? Yes, it is true. ✓
Test a value outside it: x = 1 gives left side = 3 × 1 + 5 = 8, right side = 5 × 1 + 1 = 6. Is 8 ≤ 6? No, it is false, so 1 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 < 6x + 5
2. 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:
   −3 < 4x + 5
3. Get the plain numbers on the other side. Subtract 5 from both sides.
   −8 < 4x
4. 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
5. 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
6. 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 = 2 × (−1) − 3 = −5, right side = 6 × (−1) + 5 = −1. Is −5 < −1? Yes, it is true. ✓
Test a value outside it: x = −3 gives left side = 2 × (−3) − 3 = −9, right side = 6 × (−3) + 5 = −13. Is −9 < −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.
   8x + 1 ≥ 3x − 9
2. 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:
   5x + 1 ≥ −9
3. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   5x ≥ −10
4. 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 ≥ −10 ÷ 5 = −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 = 8 × (−1) + 1 = −7, right side = 3 × (−1) − 9 = −12. Is −7 ≥ −12? Yes, it is true. ✓
Test a value outside it: x = −3 gives left side = 8 × (−3) + 1 = −23, right side = 3 × (−3) − 9 = −18. Is −23 ≥ −18? 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.
   3(x + 2) > 15
2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   3 × x = 3x
   3 × 2 = 6
   3x + 6 > 15
3. Get the plain numbers on the other side. Subtract 6 from both sides. This is the inverse of adding 6.
   3x > 9
4. 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
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 = 3 × (4 + 2) = 18, right side = 15. Is 18 > 15? Yes, it is true. ✓
Test a value outside it: x = 2 gives left side = 3 × (2 + 2) = 12, right side = 15. Is 12 > 15? 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.
   2(x − 3) ≤ 10
2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   2 × x = 2x
   2 × (−3) = −6
   2x − 6 ≤ 10
3. Get the plain numbers on the other side. Add 6 to both sides. This is the inverse of subtracting 6.
   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 − 3) = 8, right side = 10. Is 8 ≤ 10? Yes, it is true. ✓
Test a value outside it: x = 9 gives left side = 2 × (9 − 3) = 12, right side = 10. Is 12 ≤ 10? 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.
   4(2x + 1) < 36
2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   4 × 2x = 8x
   4 × 1 = 4
   8x + 4 < 36
3. Get the plain numbers on the other side. Subtract 4 from both sides. This is the inverse of adding 4.
   8x < 32
4. Divide both sides by 8. 8x means 8 lots of x. The inverse of multiplying by 8 is dividing by 8. 8 is positive, so the inequality sign stays the same.
   x < 32 ÷ 8 = 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 = 4 × (2 × 3 + 1) = 28, right side = 36. Is 28 < 36? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 4 × (2 × 5 + 1) = 44, right side = 36. Is 44 < 36? 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 ≥ 2(x + 6)
2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   2 × x = 2x
   2 × 6 = 12
   5x ≥ 2x + 12
3. 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:
   3x ≥ 12
4. 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
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 = 5 gives left side = 5 × 5 = 25, right side = 2 × (5 + 6) = 22. Is 25 ≥ 22? Yes, it is true. ✓
Test a value outside it: x = 3 gives left side = 5 × 3 = 15, right side = 2 × (3 + 6) = 18. Is 15 ≥ 18? 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.
   2(3x − 1) > 4(x + 2)
2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   2 × 3x = 6x
   2 × (−1) = −2
   4 × x = 4x
   4 × 2 = 8
   6x − 2 > 4x + 8
3. 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:
   2x − 2 > 8
4. Get the plain numbers on the other side. Add 2 to both sides. This is the inverse of subtracting 2.
   2x > 10
5. 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
6. 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 = 2 × (3 × 6 − 1) = 34, right side = 4 × (6 + 2) = 32. Is 34 > 32? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 2 × (3 × 4 − 1) = 22, right side = 4 × (4 + 2) = 24. Is 22 > 24? 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.
   5(x − 2) ≥ 3(x + 4)
2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
   5 × x = 5x
   5 × (−2) = −10
   3 × x = 3x
   3 × 4 = 12
   5x − 10 ≥ 3x + 12
3. 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:
   2x − 10 ≥ 12
4. Get the plain numbers on the other side. Add 10 to both sides. This is the inverse of subtracting 10.
   2x ≥ 22
5. 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 ≥ 22 ÷ 2 = 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 = 12 gives left side = 5 × (12 − 2) = 50, right side = 3 × (12 + 4) = 48. Is 50 ≥ 48? Yes, it is true. ✓
Test a value outside it: x = 10 gives left side = 5 × (10 − 2) = 40, right side = 3 × (10 + 4) = 42. Is 40 ≥ 42? 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 + 12 > 3
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 + 1 > 6
3. Get the plain numbers on the other side. Subtract 1 from both sides. This is the inverse of adding 1.
   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 + 12 = 3.5, right side = 3. Is 3.5 > 3? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 4 + 12 = 2.5, right side = 3. Is 2.5 > 3? 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 < −1
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 < −5
3. Get the plain numbers on the other side. Subtract 4 from both sides. This is the inverse of adding 4.
   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 = (−10) + 45 = −1.2, right side = −1. Is −1.2 < −1? Yes, it is true. ✓
Test a value outside it: x = −8 gives left side = (−8) + 45 = −0.8, right side = −1. Is −0.8 < −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 ≥ 1
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 ≥ 2
3. Get the plain numbers on the other side. Add 6 to both sides. This is the inverse of subtracting 6.
   x ≥ 8
4. 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 = 9 gives left side = 92 − 3 = 1.5, right side = 1. Is 1.5 ≥ 1? Yes, it is true. ✓
Test a value outside it: x = 7 gives left side = 72 − 3 = 0.5, right side = 1. Is 0.5 ≥ 1? 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.