Year 10 Maths · Unit 2 · Lesson 11

Linear Inequalities

Solve inequalities with confidence, graph them on a number line, and master interval notation.

35 min 8 cards 130 XP
x -1 0 1 3 x <= 3
Think First
warm-up

Before we begin: If $3 < 5$, what happens if you multiply both sides by $-1$? Is $-3 < -5$ true? Explain why or why not.

Record your answer in your workbook.
1
The Big Idea
+5 XP to read

An inequality shows that one expression is greater than or less than another. You solve inequalities almost exactly like equations, with one critical difference: when you multiply or divide both sides by a negative number, you must reverse the inequality sign.

$3 < 5$ becomes $-3 > -5$ when multiplied by $-1$.

3 < 5 x (-1) -3 > -5
flip the sign
2
Lesson Objectives
what you will learn
  • Solve linear inequalities using inverse operations
  • Reverse the inequality sign when multiplying or dividing by a negative
  • Graph solutions on a number line using open and closed circles
  • Write solutions in interval notation
  • Solve inequalities with brackets and variables on both sides
3
Key Vocabulary
terms to know
InequalityA statement that one expression is greater than, less than, or not equal to another, e.g. x > 3.
Number lineA visual representation of numbers as points on a line, used to show inequality solutions.
Interval notationA way to write continuous sets of numbers: (a, b) for open, [a, b] for closed.
Open circleUsed for < and > to show the endpoint is not included.
Closed circleUsed for ≤ and ≥ to show the endpoint is included.
Compound inequalityTwo inequalities joined together, e.g. 1 < x ≤ 5.
4
Spot the Trap
heads-up

Wrong: Forgetting to reverse the inequality when dividing by a negative. $-2x > 6$ becoming $x > -3$.

Right: When dividing by $-2$, reverse the sign: $x < -3$.

Wrong: Using a closed circle for $<$ or $>$ on the number line.

Right: Open circle for $<$ and $>$; closed circle for $\\leq$ and $\\geq$.

5
Solving Linear Inequalities
+5 XP

You can add or subtract any number from both sides of an inequality without changing the direction. You can also multiply or divide by a positive number safely. But when you multiply or divide by a negative number, you must reverse the inequality sign.

$-2x > 6$: divide by $-2$ and flip the sign. $x < -3$.

-2x > 6 / (-2), flip x < -3
flip when negative
Check the multiplier
Before dividing, look at the sign. If it is negative, write FLIP as a reminder.
Why it flips
Multiplying by -1 reflects numbers across zero. The order reverses.
Test a value
Pick a number in your solution and check it satisfies the original inequality.
6
Graphing on a Number Line
+5 XP

A number line shows all the values that satisfy an inequality. Use an open circle for < and > (the endpoint is not included). Use a closed circle for ≤ and ≥ (the endpoint is included). Shade in the direction of all valid solutions.

x > -1: open circle at -1, shade right.

-1 0 x > -1
open for < >
Open vs closed
< and > mean the endpoint is excluded. ≤ and ≥ mean it is included.
Shade the solution
All numbers that make the inequality true get shaded. Think: which side works?
Test the endpoint
Plug the boundary value back in. If it makes the statement true, use a closed circle.
7
Brackets and Both Sides
+5 XP

When an inequality contains brackets, expand them first. When the variable appears on both sides, move all variable terms to one side and constants to the other. Then isolate as usual. Remember to flip the sign if you divide by a negative.

$3(x-2) < 2(x+4)$: expand, then gather x, then isolate.

3(x-2) < 2(x+4) 3x - 6 < 2x + 8 x < 14
expand first
Distribute carefully
3(x - 2) = 3x - 6. Do not forget to multiply the constant term.
Subtract the smaller coefficient
To avoid negatives, subtract 2x from 3x, not the other way around.
Check with a test value
Try x = 0: 3(-2) = -6 and 2(4) = 8. -6 < 8 is true. The solution is correct.
8
Interval Notation
+5 XP

Interval notation is a compact way to write the set of all numbers that satisfy an inequality. A round bracket ( ) means the endpoint is not included. A square bracket [ ] means the endpoint is included. Infinity always uses a round bracket.

x ≤ 4 becomes (-∞, 4]. Square bracket because 4 is included.

x <= 4 (-inf, 4] [ means included
round = open, square = closed
Match brackets to circles
Round brackets match open circles. Square brackets match closed circles.
Infinity is always round
You can never reach infinity, so it always uses a round bracket.
Write smallest first
Always write the smaller number on the left: (-∞, 5] not [5, -∞).

Watch Me Solve It · 3 examples

Watch Me Solve It · Basic inequality
+15 XP per step
Q1
PROBLEM
Solve $3x - 5 < 10$ and write the solution in interval notation.
  1. 1
    Add 5 to both sides
    $3x - 5 + 5 < 10 + 5$
    Isolating the term with x. The inequality direction stays the same.
  2. 2
    Simplify and divide by 3
    $3x < 15$  →  $\\frac{3x}{3} < \\frac{15}{3}$
    Dividing by a positive number, so the sign does not flip.
  3. 3
    Write the solution and check
    $x < 5$  →  interval: $(-\\infty, 5)$
    Test x = 4: 3(4) - 5 = 7 < 10. True. Round bracket because 5 is not included.
Answer$x < 5$ or $(-\infty, 5)$
Watch Me Solve It · Negative coefficient
+15 XP per step
Q2
PROBLEM
Solve $-2x + 8 \\geq 4$ and graph the solution on a number line.
  1. 1
    Subtract 8 from both sides
    $-2x + 8 - 8 \\geq 4 - 8$
    Getting the variable term alone on the left.
  2. 2
    Simplify and divide by -2
    $-2x \\geq -4$  →  $\\frac{-2x}{-2} \\leq \\frac{-4}{-2}$
    Critical: Dividing by a negative number, so the sign FLIPS from ≥ to ≤.
  3. 3
    Write the solution and graph
    $x \\leq 2$  →  interval: $(-\\infty, 2]$
    Closed circle at 2 (because of ≤), shade to the left. Test x = 0: -2(0) + 8 = 8 ≥ 4. True.
Answer$x \leq 2$ or $(-\infty, 2]$
Watch Me Solve It · Brackets and both sides
+15 XP per step
Q3
PROBLEM
Solve $3(x - 2) < 2(x + 4)$ and describe the graph.
  1. 1
    Expand both sides
    $3(x - 2) = 3x - 6$    and    $2(x + 4) = 2x + 8$
    Distribute the 3 and the 2 across every term inside each bracket.
  2. 2
    Write the expanded inequality
    $3x - 6 < 2x + 8$
    Now we have variables on both sides. Gather them to one side.
  3. 3
    Subtract 2x and add 6
    $3x - 2x < 8 + 6$  →  $x < 14$
    Subtracting the smaller coefficient (2x) avoids negatives. No sign flip needed.
  4. 4
    Describe the graph and check
    Open circle at 14, shade to the left. Interval: $(-\\infty, 14)$
    Test x = 0: 3(-2) = -6 and 2(4) = 8. -6 < 8 is true. Correct.
Answer$x < 14$ or $(-\infty, 14)$

Brain Trainer · 4 problems

D
Brain Trainer · Inequalities
4 problems

Four inequalities ranging from basic to brackets and both sides. Show each step, then reveal the answer to check.

  1. 1 Solve $2x + 3 < 11$.

    Subtract 3: $2x < 8$. Divide by 2: $x < 4$. Interval: $(-\infty, 4)$. Check: $2(3) + 3 = 9 < 11$.

  2. 2 Solve $-3x \\geq 12$.

    Divide by $-3$ and flip: $x \leq -4$. Interval: $(-\infty, -4]$. Check: $-3(-5) = 15 \geq 12$, but $-3(-4) = 12 \geq 12$.

  3. 3 Solve $5x - 2 > 3x + 8$.

    Subtract $3x$: $2x - 2 > 8$. Add 2: $2x > 10$. Divide by 2: $x > 5$. Interval: $(5, \infty)$.

  4. 4 Solve $2(x + 1) \\leq 3x - 4$.

    Expand: $2x + 2 \leq 3x - 4$. Subtract $2x$: $2 \leq x - 4$. Add 4: $6 \leq x$ or $x \geq 6$. Interval: $[6, \infty)$.
Complete in your workbook.

Quick Check · 5 questions

1
Solve $3x - 5 < 10$.
+10 XP
2
Solve $-2x + 8 \\geq 4$.
+10 XP
3
Which number line shows $x > -1$?
+10 XP
4
The solution to $5 - 3x < 11$ is:
+10 XP
5
In interval notation, $x \\leq 4$ is written as:
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. Solve $6 - 2x \\geq 10$. Show all steps, explain why the inequality sign is reversed, and write the solution in interval notation.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. Solve $3(x - 2) < 2(x + 4)$. Show all steps including expansion and isolation. Describe how you would graph the solution on a number line.

Answer in your workbook.
Evaluate Medium 3 MARKS

Q8. A theme park ride has a minimum height requirement of 120 cm and a maximum of 195 cm. Let $h$ represent a person's height in centimetres.

(a) Write an inequality to describe the allowable heights. (1 mark)

(b) A child is 115 cm tall. Explain mathematically why they cannot ride. (1 mark)

(c) Another child grows 5 cm and now measures exactly 120 cm. Can they ride? Explain using the correct inequality symbol. (1 mark)

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B -- Add 5: $3x < 15$. Divide by 3: $x < 5$.

2. C -- Subtract 8: $-2x \geq -4$. Divide by $-2$ and flip: $x \leq 2$.

3. B -- Open circle at $-1$, shade right. $>$ means not included, larger values.

4. C -- Subtract 5: $-3x < 6$. Divide by $-3$ and flip: $x > -2$.

5. C -- $x \leq 4$ means 4 is included (square bracket) and values extend to $-\infty$ (round bracket).

Show Your Working Model Answers

Q6 (3 marks): $6 - 2x \geq 10$ → $-2x \geq 4$ [1]. Divide by $-2$ and reverse the sign: $x \leq -2$ [1]. The sign reverses because dividing by a negative reflects values across zero on the number line. Interval: $(-\infty, -2]$ [1].

Q7 (3 marks): Expand: $3x - 6 < 2x + 8$ [1]. Subtract $2x$: $x - 6 < 8$ [0.5]. Add 6: $x < 14$ [0.5]. Graph: open circle at 14, shade to the left [0.5]. Interval: $(-\infty, 14)$ [0.5].

Q8 (3 marks): (a) $120 \leq h \leq 195$ [1]. (b) $115 < 120$, so 115 does not satisfy $h \geq 120$. The child is below the minimum height requirement [1]. (c) Yes, they can ride. $h = 120$ satisfies $h \geq 120$ because the inequality uses $\geq$ (greater than or equal to). The endpoint 120 is included, as shown by the closed circle on a number line graph [1].

Stretch Challenge · +25 XP, +10 coins

The Compound Conundrum

Solve $-5 < 2x - 3 \\leq 7$ and express the solution in interval notation. Show every step, explain why you do or do not flip any signs, and describe the graph on a number line.

Reveal solution

Split into two inequalities:

Left: $-5 < 2x - 3$ → $-2 < 2x$ → $-1 < x$

Right: $2x - 3 \leq 7$ → $2x \leq 10$ → $x \leq 5$

Combine: $-1 < x \leq 5$

Interval notation: $(-1, 5]$

Graph: open circle at $-1$, closed circle at $5$, shade between.

No sign flips needed because we only divided by positive 2.

R
Quick Review

Flip the Sign

Always reverse when multiplying or dividing by a negative

Open vs Closed

Open circle for < and >; closed for ≤ and ≥

Expand First

Remove brackets before isolating the variable

Interval Notation

Round brackets for open, square for closed

Test a Value

Pick a number in your solution and check it

Same as Equations

All other solving rules are identical

Interactive: Inequality Planner

Practise solving inequalities, graphing on a number line, and writing interval notation with instant feedback.

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