GCSE Maths: Double inequalities and whole-number solutions (Higher)
Part of the GCSE Maths inequalities chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Double inequalities and whole-number solutions
A double inequality has two inequality signs, and a value in the middle that is squeezed between two limits. For example, 3 < n ≤ 8 means “n is bigger than 3 and at most 8”. The numbers that work are 4, 5, 6, 7 and 8 (if n is a whole number), but also 3.5, 7.9 and so on if it can be any number.
Integers. An integer is a whole number: positive, negative or zero (… −2, −1, 0, 1, 2 …). Many questions ask you to “write down all the integers that satisfy” an inequality. The best method is: draw a number line, mark the two limits (open or closed circles), and list the whole numbers between them. Watch the circles: an open circle leaves the end number out, a closed circle keeps it in.
Solving a double inequality. When the middle is not just a letter, such as 3 < 2x + 1 ≤ 11, we do the same operation to all three parts.
- Subtract 1 from all three parts: 2 < 2x ≤ 10.
- Divide all three parts by 2: 1 < x ≤ 5.
Notice that the signs stay as they were. The answer says x is bigger than 1 and at most 5.
Two separate inequalities. Sometimes you are given two inequalities, such as x + 3 > 5 and 2x < 14. Solve each one, then find the values that satisfy both, where the two number lines overlap.
A negative in the middle (Higher). If the middle has a negative x term, such as 7 < 3 − 2x ≤ 11, dividing by −2 reverses both signs, and you then write the answer with the smaller number first.
Key ideas
- A double inequality has a letter or expression squeezed between two limits.
- Do every step to all three parts.
- Integers are whole numbers (including negatives and zero). Use a number line to list them.
- Open end = not included; closed end = included.
- For two separate inequalities, keep only the values that satisfy both.
- Higher: dividing all three parts by a negative reverses both signs.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than −3 and smaller than or equal to 2. The word integer means a whole number (positive, negative or zero).−3 < n ≤ 2
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above −3. For the upper end: included, so up to 2.Smallest = −2, largest = 2
- List the integers from smallest to largest. Count up in ones.−2, −1, 0, 1, 2
Check. Test the end integers: −2 and 2 satisfy −3 < n ≤ 2, but −3 and 3 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.3 < 2x + 1 ≤ 11
- Subtract 1 from all three parts. This removes the plain number from the middle.2 < 2x ≤ 10
- Divide all three parts by 2. 2 is positive, so both signs stay the same.2 ÷ 2 = 1 and 10 ÷ 2 = 5
1 < x ≤ 5 - State the solution. This is the range of values the letter can take.1 < x ≤ 5
Check. Test x = 4: the middle becomes 2 × 4 + 1 = 9, and 3 < 9 ≤ 11 is true. ✓
Test the end value x = 1: the middle becomes 3, which equals the left end 3, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−4 ≤ 3x − 1 < 11
- Add 1 to all three parts. This removes the plain number from the middle.−3 ≤ 3x < 12
- Divide all three parts by 3. 3 is positive, so both signs stay the same.−3 ÷ 3 = −1 and 12 ÷ 3 = 4
−1 ≤ x < 4 - State the solution. This is the range of values the letter can take.−1 ≤ x < 4
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.−1, 0, 1, 2, 3
Check. Test x = 1: the middle becomes 3 × 1 − 1 = 2, and −4 ≤ 2 < 11 is true. ✓
Test the end value x = −1: the middle becomes −4, which equals the left end −4, so the end is included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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 each inequality on its own, then keep only the values that satisfy both (where the number lines overlap).
- Solve the first inequality. Treat each inequality separately, using the same method as for one.x + 3 > 5
gives x > 2 - Solve the second inequality. Do the same for the other one.2x < 14
gives x < 7 - Show both on one number line. The values that satisfy both inequalities are where the two lines overlap.
- Write the values that satisfy both. Take the overlap of the two ranges.2 < x < 7
- List the integers. Write down the whole numbers in the overlap.3, 4, 5, 6
Check. Pick a value in the overlap and check it in both original inequalities. ✓
Watch out. The answer must satisfy both inequalities, not just one of them.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.7 < −2x + 3 ≤ 11
- Subtract 3 from all three parts. This removes the plain number from the middle.4 < −2x ≤ 8
- Divide all three parts by −2 and reverse both signs. −2 is negative, so dividing flips the order. Both signs reverse.−2 > x ≥ −4
- Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.−4 ≤ x < −2
- State the solution. This is the range of values the letter can take.−4 ≤ x < −2
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.−4, −3
Check. Test x = −3: the middle becomes −2 × (−3) + 3 = 9, and 7 < 9 ≤ 11 is true. ✓
Test the end value x = −4: the middle becomes 11, which equals the right end 11, so the end is included.
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.
Watch out
Do not forget that an open circle leaves the end number out. For −3 < n ≤ 2 the whole numbers are −2, −1, 0, 1, 2. Many students wrongly include −3 or leave out 2. And when solving, do every step to all three parts, not just the middle.Exam tip
To list integers, draw a quick number line and put a dot on every whole number between the limits. Then double-check the two end numbers against the circles.Exercise 10F Double inequalities and integer solutions
Non-calculatorKey words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than or equal to −3 and smaller than 2. The word integer means a whole number (positive, negative or zero).−3 ≤ n < 2
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is included, so start at −3. For the upper end: not included, so the integer just below 2.Smallest = −3, largest = 1
- List the integers from smallest to largest. Count up in ones.−3, −2, −1, 0, 1
Check. Test the end integers: −3 and 1 satisfy −3 ≤ n < 2, but −4 and 2 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than 1 and smaller than or equal to 6. The word integer means a whole number (positive, negative or zero).1 < n ≤ 6
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above 1. For the upper end: included, so up to 6.Smallest = 2, largest = 6
- List the integers from smallest to largest. Count up in ones.2, 3, 4, 5, 6
Check. Test the end integers: 2 and 6 satisfy 1 < n ≤ 6, but 1 and 7 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than −5 and smaller than −1. The word integer means a whole number (positive, negative or zero).−5 < n < −1
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above −5. For the upper end: not included, so the integer just below −1.Smallest = −4, largest = −2
- List the integers from smallest to largest. Count up in ones.−4, −3, −2
Check. Test the end integers: −4 and −2 satisfy −5 < n < −1, but −5 and −1 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than or equal to 0 and smaller than or equal to 4. The word integer means a whole number (positive, negative or zero).0 ≤ n ≤ 4
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is included, so start at 0. For the upper end: included, so up to 4.Smallest = 0, largest = 4
- List the integers from smallest to largest. Count up in ones.0, 1, 2, 3, 4
Check. Test the end integers: 0 and 4 satisfy 0 ≤ n ≤ 4, but −1 and 5 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than −2.5 and smaller than or equal to 1.5. The word integer means a whole number (positive, negative or zero).−2.5 < n ≤ 1.5
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above −2.5. For the upper end: round 1.5 down.Smallest = −2, largest = 1
- List the integers from smallest to largest. Count up in ones.−2, −1, 0, 1
Check. Test the end integers: −2 and 1 satisfy −2.5 < n ≤ 1.5, but −3 and 2 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than −1 and smaller than 1. The word integer means a whole number (positive, negative or zero).−1 < n < 1
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above −1. For the upper end: not included, so the integer just below 1.Smallest = 0, largest = 0
- List the integers from smallest to largest. Count up in ones.0
Check. Test the end integers: 0 and 0 satisfy −1 < n < 1, but −1 and 1 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.4 < 2x ≤ 12
- Divide all three parts by 2. 2 is positive, so both signs stay the same.4 ÷ 2 = 2 and 12 ÷ 2 = 6
2 < x ≤ 6 - State the solution. This is the range of values the letter can take.2 < x ≤ 6
Check. Test x = 5: the middle becomes 2 × 5 = 10, and 4 < 10 ≤ 12 is true. ✓
Test the end value x = 2: the middle becomes 4, which equals the left end 4, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−3 ≤ x + 1 < 5
- Subtract 1 from all three parts. This removes the plain number from the middle.−4 ≤ x < 4
- State the solution. This is the range of values the letter can take.−4 ≤ x < 4
Check. Test x = 0: the middle becomes 0 + 1 = 1, and −3 ≤ 1 < 5 is true. ✓
Test the end value x = −4: the middle becomes −3, which equals the left end −3, so the end is included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.6 < 3x ≤ 15
- Divide all three parts by 3. 3 is positive, so both signs stay the same.6 ÷ 3 = 2 and 15 ÷ 3 = 5
2 < x ≤ 5 - State the solution. This is the range of values the letter can take.2 < x ≤ 5
Check. Test x = 4: the middle becomes 3 × 4 = 12, and 6 < 12 ≤ 15 is true. ✓
Test the end value x = 2: the middle becomes 6, which equals the left end 6, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.1 ≤ 2x − 5 < 9
- Add 5 to all three parts. This removes the plain number from the middle.6 ≤ 2x < 14
- Divide all three parts by 2. 2 is positive, so both signs stay the same.6 ÷ 2 = 3 and 14 ÷ 2 = 7
3 ≤ x < 7 - State the solution. This is the range of values the letter can take.3 ≤ x < 7
Check. Test x = 5: the middle becomes 2 × 5 − 5 = 5, and 1 ≤ 5 < 9 is true. ✓
Test the end value x = 3: the middle becomes 1, which equals the left end 1, so the end is included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.0 < 5x + 10 ≤ 40
- Subtract 10 from all three parts. This removes the plain number from the middle.−10 < 5x ≤ 30
- Divide all three parts by 5. 5 is positive, so both signs stay the same.−10 ÷ 5 = −2 and 30 ÷ 5 = 6
−2 < x ≤ 6 - State the solution. This is the range of values the letter can take.−2 < x ≤ 6
Check. Test x = 3: the middle becomes 5 × 3 + 10 = 25, and 0 < 25 ≤ 40 is true. ✓
Test the end value x = −2: the middle becomes 0, which equals the left end 0, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−7 < 2x − 1 ≤ 3
- Add 1 to all three parts. This removes the plain number from the middle.−6 < 2x ≤ 4
- Divide all three parts by 2. 2 is positive, so both signs stay the same.−6 ÷ 2 = −3 and 4 ÷ 2 = 2
−3 < x ≤ 2 - State the solution. This is the range of values the letter can take.−3 < x ≤ 2
Check. Test x = 0: the middle becomes 2 × 0 − 1 = −1, and −7 < −1 ≤ 3 is true. ✓
Test the end value x = −3: the middle becomes −7, which equals the left end −7, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.3 < 2x + 1 ≤ 11
- Subtract 1 from all three parts. This removes the plain number from the middle.2 < 2x ≤ 10
- Divide all three parts by 2. 2 is positive, so both signs stay the same.2 ÷ 2 = 1 and 10 ÷ 2 = 5
1 < x ≤ 5 - State the solution. This is the range of values the letter can take.1 < x ≤ 5
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.2, 3, 4, 5
Check. Test x = 4: the middle becomes 2 × 4 + 1 = 9, and 3 < 9 ≤ 11 is true. ✓
Test the end value x = 1: the middle becomes 3, which equals the left end 3, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−4 ≤ 3x − 1 < 11
- Add 1 to all three parts. This removes the plain number from the middle.−3 ≤ 3x < 12
- Divide all three parts by 3. 3 is positive, so both signs stay the same.−3 ÷ 3 = −1 and 12 ÷ 3 = 4
−1 ≤ x < 4 - State the solution. This is the range of values the letter can take.−1 ≤ x < 4
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.−1, 0, 1, 2, 3
Check. Test x = 1: the middle becomes 3 × 1 − 1 = 2, and −4 ≤ 2 < 11 is true. ✓
Test the end value x = −1: the middle becomes −4, which equals the left end −4, so the end is included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.2 ≤ 4x − 2 < 18
- Add 2 to all three parts. This removes the plain number from the middle.4 ≤ 4x < 20
- Divide all three parts by 4. 4 is positive, so both signs stay the same.4 ÷ 4 = 1 and 20 ÷ 4 = 5
1 ≤ x < 5 - State the solution. This is the range of values the letter can take.1 ≤ x < 5
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.1, 2, 3, 4
Check. Test x = 3: the middle becomes 4 × 3 − 2 = 10, and 2 ≤ 10 < 18 is true. ✓
Test the end value x = 1: the middle becomes 2, which equals the left end 2, so the end is included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−6 < 2x ≤ 6
- Divide all three parts by 2. 2 is positive, so both signs stay the same.−6 ÷ 2 = −3 and 6 ÷ 2 = 3
−3 < x ≤ 3 - State the solution. This is the range of values the letter can take.−3 < x ≤ 3
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.−2, −1, 0, 1, 2, 3
Check. Test x = 1: the middle becomes 2 × 1 = 2, and −6 < 2 ≤ 6 is true. ✓
Test the end value x = −3: the middle becomes −6, which equals the left end −6, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
Method. List the whole numbers, then count them.
- Draw the number line. Open circle at −2 (not included), closed circle at 3 (included).
- List the whole numbers. Start at the first one after −2 and finish at 3.−1, 0, 1, 2, 3
- Count them. 5
Check. 3 − (−2) = 5, and when one end is included and the other is not, the number of integers equals that difference. ✓
Watch out. Do not count the end that has an open circle.
Method. Read the dots, then use the circles to decide the signs.
- Read the dots. Each dot is a whole number that works.−3, −2, −1, 0, 1
- Look at the circles. A closed circle at −3 (included) and an open circle at 2 (not included).−3 is included, 2 is not
- Write the inequality. Closed circle gives ≤, open circle gives <.−3 ≤ n < 2
Check. List the integers satisfying the inequality: −3, −2, −1, 0, 1 — matching the dots. ✓
Watch out. Open circle: not included.
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 each inequality on its own, then keep only the values that satisfy both (where the number lines overlap).
- Solve the first inequality. Treat each inequality separately, using the same method as for one.2x − 1 > 5
gives x > 3 - Solve the second inequality. Do the same for the other one.x < 7
gives x < 7 - Show both on one number line. The values that satisfy both inequalities are where the two lines overlap.
- Write the values that satisfy both. Take the overlap of the two ranges.3 < x < 7
- List the integers. Write down the whole numbers in the overlap.4, 5, 6
Check. Pick a value in the overlap and check it in both original inequalities. ✓
Watch out. The answer must satisfy both inequalities, not just one of them.
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 each inequality on its own, then keep only the values that satisfy both (where the number lines overlap).
- Solve the first inequality. Treat each inequality separately, using the same method as for one.3x − 2 ≥ 4
gives x ≥ 2 - Solve the second inequality. Do the same for the other one.2x + 1 < 11
gives x < 5 - Show both on one number line. The values that satisfy both inequalities are where the two lines overlap.
- Write the values that satisfy both. Take the overlap of the two ranges.2 ≤ x < 5
- List the integers. Write down the whole numbers in the overlap.2, 3, 4
Check. Pick a value in the overlap and check it in both original inequalities. ✓
Watch out. The answer must satisfy both inequalities, not just one of them.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.24 < 4x + 8 ≤ 40
- Subtract 8 from all three parts. This removes the plain number from the middle.16 < 4x ≤ 32
- Divide all three parts by 4. 4 is positive, so both signs stay the same.16 ÷ 4 = 4 and 32 ÷ 4 = 8
4 < x ≤ 8 - State the solution. This is the range of values the letter can take.4 < x ≤ 8
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.5, 6, 7, 8
Check. Test x = 7: the middle becomes 4 × 7 + 8 = 36, and 24 < 36 ≤ 40 is true. ✓
Test the end value x = 4: the middle becomes 24, which equals the left end 24, so the end is not included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.9 ≤ 2m + 3 ≤ 21
- Subtract 3 from all three parts. This removes the plain number from the middle.6 ≤ 2m ≤ 18
- Divide all three parts by 2. 2 is positive, so both signs stay the same.6 ÷ 2 = 3 and 18 ÷ 2 = 9
3 ≤ m ≤ 9 - State the solution. This is the range of values the letter can take.3 ≤ m ≤ 9
- List the integers (whole numbers) m can be. Use the number line to see which whole numbers fit inside the range.3, 4, 5, 6, 7, 8, 9
Check. Test m = 6: the middle becomes 2 × 6 + 3 = 15, and 9 ≤ 15 ≤ 21 is true. ✓
Test the end value m = 3: the middle becomes 9, which equals the left end 9, so the end is included.
Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.
Method. Solve the double inequality, then count the whole numbers.
- Solve it. Subtract 1 from all three parts, then divide all three by 2.−4 < x ≤ 4
- List the integers. Open circle at −4, closed at 4.−3, −2, −1, 0, 1, 2, 3, 4
- Count them. 8
Check. Test x = -3: 2 × (-3) + 1 = -5, and −7 < -5 ≤ 9. ✓
Watch out. −4 is not included (open circle).
Method. Solve it correctly, then compare.
- Solve it correctly. To remove the −1 from the middle we add 1 to all three parts.3 < 3x < 12
- Divide by 3. All three parts.1 < x < 4
- Find Ali’s mistake. He subtracted 1 from the outer numbers instead of adding 1.2 − 1 = 1 and 11 − 1 = 10 (wrong)
Check. Test x = 2 in 2 < 3x − 1 < 11: 3 × 2 − 1 = 5, and 2 < 5 < 11 is true. ✓
Watch out. To undo “minus 1”, you add 1.
Method. List the whole numbers near the limits and test them.
- Test −1 and 1. Both ends are open circles, so −1 and 1 are not allowed.−1 < −1 is false; 1 < 1 is false
- Test 0. Is −1 < 0 < 1?−1 < 0 and 0 < 1 are both true
- Conclude. There are no other whole numbers between −1 and 1.The only integer is 0.
Check. If the ends were included (−1 ≤ n ≤ 1), the integers would be −1, 0, 1. ✓
Watch out. Check the end numbers against the circles.
Method. Work out what each condition allows, then keep the numbers that satisfy both.
- First condition. The squares of whole numbers: 0, 1, 4, 9, 16, 25, 36. Squares below 30 come from 0, ±1, ±2, ±3, ±4, ±5.−5 ≤ n ≤ 5
- Second condition. Subtract 3, then divide by 2.2n ≥ −8, so n ≥ −4
- Keep the numbers that satisfy both. n is from −4 to 5.−4, −3, −2, −1, 0, 1, 2, 3, 4, 5
Check. −5: 2 × (−5) + 3 = −7, which is not ≥ −5, so −5 is out. 5: 25 < 30 and 13 ≥ −5. ✓
Watch out. List each condition separately, then combine.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.7 < −2x + 3 ≤ 11
- Subtract 3 from all three parts. This removes the plain number from the middle.4 < −2x ≤ 8
- Divide all three parts by −2 and reverse both signs. −2 is negative, so dividing flips the order. Both signs reverse.−2 > x ≥ −4
- Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.−4 ≤ x < −2
- State the solution. This is the range of values the letter can take.−4 ≤ x < −2
Check. Test x = −3: the middle becomes −2 × (−3) + 3 = 9, and 7 < 9 ≤ 11 is true. ✓
Test the end value x = −4: the middle becomes 11, which equals the right end 11, so the end is included.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−4 ≤ −3x + 5 < 8
- Subtract 5 from all three parts. This removes the plain number from the middle.−9 ≤ −3x < 3
- Divide all three parts by −3 and reverse both signs. −3 is negative, so dividing flips the order. Both signs reverse.3 ≥ x > −1
- Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.−1 < x ≤ 3
- State the solution. This is the range of values the letter can take.−1 < x ≤ 3
Check. Test x = 2: the middle becomes −3 × 2 + 5 = −1, and −4 ≤ −1 < 8 is true. ✓
Test the end value x = −1: the middle becomes 8, which equals the right end 8, so the end is not included.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.2 < −x + 1 ≤ 6
- Subtract 1 from all three parts. This removes the plain number from the middle.1 < −x ≤ 5
- Divide all three parts by −1 and reverse both signs. −1 is negative, so dividing flips the order. Both signs reverse.−1 > x ≥ −5
- Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.−5 ≤ x < −1
- State the solution. This is the range of values the letter can take.−5 ≤ x < −1
Check. Test x = −3: the middle becomes −(−3) + 1 = 4, and 2 < 4 ≤ 6 is true. ✓
Test the end value x = −5: the middle becomes 6, which equals the right end 6, so the end is included.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−9 ≤ −4x + 3 < 0
- Subtract 3 from all three parts. This removes the plain number from the middle.−12 ≤ −4x < −3
- Divide all three parts by −4 and reverse both signs. −4 is negative, so dividing flips the order. Both signs reverse.3 ≥ x > 0.75
- Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.0.75 < x ≤ 3
- State the solution. This is the range of values the letter can take.0.75 < x ≤ 3
Check. Test x = 2: the middle becomes −4 × 2 + 3 = −5, and −9 ≤ −5 < 0 is true. ✓
Test the end value x = 0.75: the middle becomes 0, which equals the right end 0, so the end is not included.
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. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.
- Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.−3 ≤ −2x + 1 < 9
- Subtract 1 from all three parts. This removes the plain number from the middle.−4 ≤ −2x < 8
- Divide all three parts by −2 and reverse both signs. −2 is negative, so dividing flips the order. Both signs reverse.2 ≥ x > −4
- Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.−4 < x ≤ 2
- State the solution. This is the range of values the letter can take.−4 < x ≤ 2
- List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.−3, −2, −1, 0, 1, 2
Check. Test x = 0: the middle becomes −2 × 0 + 1 = 1, and −3 ≤ 1 < 9 is true. ✓
Test the end value x = −4: the middle becomes 9, which equals the right end 9, so the end is not included.
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.