Year 9 Maths The Thai Guy

Lesson 8.3 · 30 min

Linear Inequalities

Solving inequalities like equations, reversing the sign when multiplying or dividing by a negative, and showing solutions on a number line.

In this lesson you will learn to…

  • Solve a linear inequality using the balance method
  • Reverse the inequality sign when multiplying or dividing by a negative number
  • Show a solution on a number line using open and closed circles
  • Solve a combined inequality such as and list integer solutions

Why this matters

Real limits are rarely exact: at least 55 cm tall, no more than 20 kg of luggage, between 2 and 7 hours. Inequalities describe a whole range of acceptable answers, and solving them tells you exactly where that range starts and stops.

Why the sign flips

Start with something true: 5>35 > 3. Multiply both sides by −1-1 and you get −5-5 and −3-3 — but −5-5 is less than −3-3. Multiplying by a negative mirrors the number line, so the order of the two sides swaps and the sign must flip to stay true.

Ashleigh solved 5−2x≤115 - 2x \le 11 and forgot:

−2x≤6  ⇒  x≤−3(wrong!)-2x \le 6 \;\Rightarrow\; x \le -3 \quad \text{(wrong!)}

The correct answer is x≥−3x \ge -3. Test x=−4x = -4 in the original: 5−2(−4)=135 - 2(-4) = 13, which is not ≤11\le 11 — so x≤−3x \le -3 cannot be right.

Reading number lines

InequalityCircleArrow
x<−5x < -5open ○ at −5-5left
x≥−2x \ge -2closed ● at −2-2right
2≤x<72 \le x < 7● at 2, ○ at 7a segment between them

Solution sets

If xx must be a whole number from 1 to 9 and x+1>7x + 1 > 7, the solution set is {7,8,9}\{7, 8, 9\} — the values that make the inequality true.

Key rules and formulas

  • The four symbols

    less than, greater than, less than or equal to, greater than or equal to. The narrow end points at the smaller value: and say the same thing.

  • Solve like an equation

    You can add, subtract, multiply or divide both sides by the same positive number and the inequality stays true.

  • The one new rule

    When you multiply or divide both sides by a negative number, reverse the inequality sign. ( but .)

  • Number lines

    A closed circle ● means the value is included ( or ). An open circle ○ means it is not ( or ). An arrow shows the direction of the solution.
Try it

Show an inequality

-2
Inequality shown
x ≥ -2

Move the point, switch the arrow direction and choose an open or closed circle. The readout shows the inequality you have drawn. Try making , then .

Worked examples

Worked example 1

Solving like an equation

Solve .

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Dividing by a negative

Solve .

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A combined inequality

Solve and list the integers that satisfy it.

Try it yourself first, then reveal the solution one step at a time.

Practice: linear inequalities

11 questions

Eleven questions. Type inequality answers like x > 3 or x <= 2 (use <= and >= for ≤ and ≥).
  1. Question 1 of 11Complete the tableFoundation
    Solve each inequality. Type the answer like x > 3.
    InequalitySolution
    5x + 2 > 17
    4x + 3 ≤ 15
    5x + 2 ≥ x + 8
    7x + 4 < 2x + 19
  2. Question 2 of 11Explore and match
    Show the solution of on the number line.

    Move the controls until you match the question, then press Check.

    0
    Inequality shown
    x > 0
  3. Question 3 of 11True or false
    When you divide both sides of an inequality by a negative number, you must reverse the inequality sign.
  4. Question 4 of 11Multiple choice
    Ashleigh solved and got . What is the correct solution?
  5. Question 5 of 11Complete the tableChallenge
    These involve negatives. Solve each one.
    InequalitySolution
    −x > 2
    −2x ≥ 8
    4 − 5x < 34
    55 ≤ 22 − 11x
  6. Question 6 of 11Explore and match
    Show on the number line.

    Move the controls until you match the question, then press Check.

    0
    Inequality shown
    x > 0
  7. Question 7 of 11Fill the gaps
    Solve the combined inequality .

  8. Question 8 of 11Choose all that apply
    Which whole numbers from 1 to 9 satisfy ?

    Select every correct answer — there may be more than one.

  9. Question 9 of 11Multiple choice
    Anya's plant is 20 cm tall and grows 0.5 cm a day. She wants it to be at least 55 cm. Which inequality shows this after days?
  10. Question 10 of 11Choose all that apply
    Start with . Which of these keep it true?

    Select every correct answer — there may be more than one.

  11. Question 11 of 11Step by stepChallenge
    Work it out step by step.
    A triangle has sides , and 25 cm, and 25 cm is the longest side. The two shorter sides must add up to more than the longest side.
    1. From , must be greater than …