Practice Maths

Linear Inequalities

Key Ideas

Key Terms

Solve like an equation
Using inverse operations — with one critical exception:
Flip (reverse) the inequality sign
When you multiply or divide both sides by a negative number.
Number line
Open circle for strict < or >; closed circle for ≤ or ≥.
Number Line Notation
SymbolCircle typeArrow direction (e.g. x > 3)
x > 3Open circle at 3Arrow pointing right
x < 3Open circle at 3Arrow pointing left
x ≥ 3Closed (filled) circle at 3Arrow pointing right
x ≤ 3Closed (filled) circle at 3Arrow pointing left
Hot Tip The most common error is forgetting to flip the inequality sign when dividing or multiplying by a negative. A good habit is to rearrange so the coefficient of x is positive before dividing.

Worked Example

Question: Solve −3x + 6 > 0.

Step 1 — Subtract 6: −3x > −6

Step 2 — Divide by −3. Flip the sign!
x < 2

Graph: Open circle at 2, arrow pointing left.

Inequalities vs Equations

An equation says two expressions are equal (=) and usually has one solution. An inequality says one expression is greater than, less than, greater than or equal to, or less than or equal to another (>, <, ≥, ≤), and typically has infinitely many solutions (a range of values).

For example, x > 3 means x can be 4, 5, 3.1, 100, or any value greater than 3. There are infinitely many solutions.

Solving Linear Inequalities

Solve inequalities using the same steps as equations — with one crucial exception: when you multiply or divide both sides by a negative number, flip the inequality sign.

Solve 2x − 3 < 7: add 3 → 2x < 10 → divide by 2 → x < 5. Sign stays the same (divided by positive 2).

Solve −3x ≥ 12: divide both sides by −3 and flip the sign → x ≤ −4. Sign flips because we divided by a negative.

Solve 4 − 2x > 10: subtract 4 → −2x > 6 → divide by −2 and flip → x < −3.

Representing Solutions on a Number Line

Draw a number line and mark the boundary value. Use a closed circle (solid dot) if the boundary is included (≥ or ≤), or an open circle if it is not (> or <). Shade the arrow in the direction of the solution.

x > 3: open circle at 3, arrow pointing right. x ≤ −4: closed circle at −4, arrow pointing left.

Interval Notation

Interval notation is a concise way to write the solution set. Use a square bracket [ or ] when the endpoint is included, and a round bracket ( or ) when it is not.

x > 3 in interval notation: (3, ∞). x ≤ −4: (−∞, −4]. Note that ∞ and −∞ always use round brackets because infinity is never actually reached.

The golden rule: The sign flip when dividing by a negative is the single most common source of errors in this topic. A good way to remember it: think about −1 < 1 (true). Multiply both sides by −1: if you do not flip, you get 1 < −1, which is false. You must flip to get 1 > −1, which is true. If you are ever unsure, test a specific number in your answer to check.

Mastery Practice

  1. Solve each inequality and describe the number line graph (circle type and arrow direction). Fluency

     InequalitySolution & graph description
    (a)x + 4 > 9 
    (b)y − 3 ≤ 5 
    (c)2x < 14 
    (d)3m ≥ −12 
    (e)5 − x < 2 
    (f)−2y > 8 
    (g)−4x ≤ 20 
    (h)4 − 2x ≥ −6 

  2. Solve each two-step inequality. State whether the sign flips and why. Fluency

     InequalitySolution (sign flip: yes/no)
    (a)2x + 3 > 11 
    (b)3y − 7 ≤ 14 
    (c)−5m + 4 ≥ −6 
    (d)6 − 3x < 0 
    (e)−2k + 10 > −4 
    (f)4(x − 1) ≤ 12 
    (g)−3(2p + 1) ≥ 9 
    (h)5 − (x + 2) > 1 

  3. Match each number line description to the correct inequality. Write the letter of the matching inequality. Fluency

    Number line descriptions:

    1. Open circle at 4, arrow pointing right
    2. Closed circle at −2, arrow pointing left
    3. Open circle at 3, arrow pointing left
    4. Closed circle at 5, arrow pointing right
    5. Open circle at 0, arrow pointing right
    6. Closed circle at −1, arrow pointing right
    7. Open circle at 7, arrow pointing left
    8. Closed circle at 6, arrow pointing left

    Inequalities to match:   A: x > 4    B: x ≤ −2    C: x < 3    D: x ≥ 5    E: x > 0    F: x ≥ −1    G: x < 7    H: x ≤ 6

  4. True or False? Fluency

    1. When you divide both sides of −4x > 12 by −4, the result is x > −3.
    2. x = 5 satisfies the inequality 3x − 4 > 11.
    3. Adding a negative number to both sides of an inequality always reverses the sign.
    4. x < 3 and x ≤ 3 have different solution sets.
    5. x = −5 is in the solution set of −4x ≤ 20.
    6. Solving 2(x + 3) ≤ 14 gives x ≤ 4.

  5. Minimum score needed. Understanding

    Test average. A student needs to average at least 70% across 4 tests to pass a unit. They scored 65, 72, and 68 on the first three tests.
    1. Let s be the score on the fourth test. Write an inequality for the average being at least 70.
    2. Solve for s. What is the minimum score needed?

  6. Taxi fare. Understanding

    Budget travel. A taxi charges a flat fee of $3.50 plus $2.20 per kilometre. Mia has at most $25 to spend.
    1. Write an inequality for Mia’s total fare being within budget.
    2. Solve and state the maximum number of whole kilometres she can travel.

  7. Inequalities with variables on both sides. Understanding

    Solving and reasoning. Solve each inequality, then give two integer values that satisfy it.
    1. 3x + 2 > x + 10
    2. 5y − 4 ≤ 2y + 8
    3. 4m − 3 ≥ 7m + 9

  8. Compound inequality. Understanding

    Between two values. A compound inequality gives both a lower and upper bound on x.
    1. Solve 1 ≤ 2x − 3 < 7 by treating the two parts together.
    2. Describe the graph: what type of circles and where?
    3. List all integer values of x that satisfy this compound inequality.

  9. Consecutive integers. Problem Solving

    Integer constraint. The sum of three consecutive integers is less than 45. Let the smallest integer be n.
    1. Write and solve an inequality to find all valid values of n.
    2. State the largest possible set of three consecutive integers satisfying the condition.
    3. Explain why you need to state that n is an integer (not just any real number).

  10. Rectangle dimensions. Problem Solving

    Perimeter constraint. The length of a rectangle is 3 cm more than twice its width. The perimeter must be at most 54 cm.
    1. Let w be the width. Write expressions for the length and perimeter, then form an inequality.
    2. Solve for the maximum width.
    3. Find the corresponding maximum length and verify the perimeter constraint.
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