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5.5 More Inequalities

GCSE Maths Foundation · Edexcel · Chapter 5: Equations, Inequalities and Sequences

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: inequalities with brackets, unknowns on both sides, double inequalities and negative terms. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Solve inequalities with brackets
  • Solve inequalities with the unknown on both sides
  • Solve double inequalities such as −3 ≤ 2x + 1 < 7 and list the integer solutions
  • Deal safely with a negative x term

Key words

  • Double inequality: Two inequalities written together, e.g. −2 ≤ x x ≥ −2 and x < 3.
  • Integer solutions: The whole numbers that make an inequality true.
  • Solution set: All the values that make the inequality true.
  • Satisfy: A number satisfies an inequality if it makes it true.

Inequalities with brackets

Treat them just like equations: expand the bracket (or divide both sides by the number outside), then solve.

Worked example: divide first

Solve 3(x − 2) < 12.
Divide both sides by 3: x − 2 < 4
Add 2: x < 6

Answer: x < 6

Worked example: expand first

Solve 2(3x + 5) ≥ 4.
Expand: 6x + 10 ≥ 4. Subtract 10: 6x ≥ −6. Divide by 6: x ≥ −1

Answer: x ≥ −1

The unknown on both sides

Worked example: subtract the smaller x term

Solve 5x + 1 > 2x + 13.
Subtract 2x: 3x + 1 > 13. Subtract 1: 3x > 12. Divide by 3: x > 4

Answer: x > 4

Worked example: a fraction answer

Solve 7x − 3 ≤ 3x + 7.
Subtract 3x: 4x − 3 ≤ 7. Add 3: 4x ≤ 10. Divide by 4: x ≤ 2.5

Answer: x ≤ 2.5

Double inequalities

A double inequality has three parts. Do the same thing to all three parts until x is on its own in the middle.

Worked example: solve and list the integers

Solve −3 ≤ 2x + 1 < 7 and list the integer solutions.
Subtract 1 from all three parts: −4 ≤ 2x < 6
Divide all three parts by 2: −2 ≤ x < 3
Integers: −2 is included, 3 is not.

Answer: −2 ≤ x < 3; integers −2, −1, 0, 1, 2

Key fact: Two separate facts can be joined: x > −1 and x ≤ 4 together give −1 x ≤ 4.

Negative x terms

If the x term is negative, add it to both sides so it becomes positive. This avoids dividing by a negative number.

Worked example: make the x term positive

Solve 10 − 2x > 4.
Add 2x to both sides: 10 > 4 + 2x
Subtract 4: 6 > 2x. Divide by 2: 3 > x, which is x < 3
Check with x = 0: 10 − 0 = 10, and 10 > 4 ✓

Answer: x < 3

Key fact: If you multiply or divide both sides by a negative number, you must reverse the inequality sign. −2x > −6 gives x < 3.

Worked example: in context

Ana has £50 and saves £8 a week. Ben has £90 and saves £3 a week. After how many whole weeks will Ana have more than Ben?
50 + 8w > 90 + 3w. 5w > 40. w > 8

Answer: after 9 weeks

Common mistakes

  • Doing something to only two of the three parts of a double inequality.
  • Dividing by a negative number and forgetting to reverse the sign.
  • Reading 3 > x as x > 3. It means x < 3.
  • Giving w = 8 when w > 8. For whole weeks the answer is 9.

Revision questions

Try these, then check your answers below each one.

Question 1. Solve 5(x - 3) > 10. (Choose from: x > 5, x > 2.6, x > -1, x < 5.)

Answer: x > 5

Working: x - 3 > 2, so x > 5.

Question 2. Solve 7x - 4 ≤ 3x + 18. What is the largest integer value of x?

Answer: 5

Working: 4x ≤ 22, x ≤ 5.5. Largest integer: 5.

Question 3. How many integers satisfy -7 ≤ 3x - 1 < 8?

Answer: 5

Working: -6 ≤ 3x < 9, -2 ≤ x < 3: -2, -1, 0, 1, 2, which is 5 integers.

Related lessons

Continue in the full course

This lesson is section 5.5 (EGCMF0505) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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