An equation pins \(x\) to a single value; an inequality allows a whole stretch of the number line. Learn to read and draw that stretch, then solve inequalities almost exactly like equations.
Solve an inequality like an equation: do the same thing to every side. The one extra rule: when you multiply or divide by a NEGATIVE number, reverse the inequality sign.
Fig. 1: −3 is included (closed circle) and 1 is not (open circle); every number between them is allowed
Section 01
Inequalities on a number line.
Set an inequality in the lab and watch the number line: the circle at each end fills in or empties as you switch between \(<\) and \(\le\). Then drag an end point and read off which whole numbers are still inside the range.
Symbol
Meaning
Circle
\(x < 3\)
less than 3
open at 3
\(x \le 3\)
less than or equal to 3
closed at 3
\(x > -1\)
greater than \(-1\)
open at \(-1\)
\(x \ge -1\)
greater than or equal to \(-1\)
closed at \(-1\)
Worked example Draw an inequality
Show \(-2 < x \le 3\) on a number line.
Left end: \(-2 < x\) is strict, so draw an OPEN circle at \(-2\).
Right end: \(x \le 3\) includes 3, so draw a CLOSED circle at 3.
Join the two circles with a line: every number between them satisfies the inequality.
Answer: Open circle at \(-2\), closed circle at 3, joined by a line.
Worked example Read an inequality
A number line shows a closed circle at \(-1\) with an arrow going to the right. Write down the inequality.
The arrow to the right means all the numbers bigger than \(-1\).
The closed circle means \(-1\) itself is included, so use \(\ge\).
Answer: \(x \ge -1\)
Worked example Integer solutions
List the integers \(n\) such that \(-3 \le n < 2\).
\(-3\) is included (\(\le\)), so start at \(-3\).
2 is NOT included (\(<\)), so stop at 1.
Answer: \(-3,\ -2,\ -1,\ 0,\ 1\)
Examiner trap
Open and closed circles are not decoration. Drawing a closed circle for \(x < 3\) says 3 is allowed, and loses the mark. Strict sign, open circle.
Section 02
Solving like an equation.
(Extended) Add, subtract, multiply and divide on both sides just as you would for an equation. The sign stays the same, with one exception.
Worked example One step at a time (Extended)
Solve \(3x - 5 < 16\).
Add 5 to both sides: \(3x < 21\).
Divide both sides by 3 (positive, so the sign stays): \(x < 7\).
Answer: \(x < 7\)
Worked example Unknown on both sides (Extended)
Solve \(3x < 2x + 4\).
Subtract \(2x\) from both sides: \(x < 4\).
Answer: \(x < 4\)
Worked example Dividing by a negative (Extended)
Solve \(5 - 2x \ge 11\).
Subtract 5 from both sides: \(-2x \ge 6\).
Divide both sides by \(-2\). Dividing by a negative REVERSES the sign: \(x \le -3\).
Check with \(x = -4\): \(5 - 2(-4) = 13 \ge 11\). True, so the direction is right.
Why the flip? \(2 < 5\) is true, but multiply both sides by \(-1\) and you get \(-2\) and \(-5\), where \(-2 > -5\). Forgetting to reverse the sign is the most common lost mark on this topic. You can avoid it by moving the \(x\) term to the side where it stays positive.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
A number line has an open circle at 2 and an arrow pointing left. Which inequality does it show?
Quick check+10 XP
List the integers \(n\) such that \(-2 < n \le 1\).
Quick check+10 XP
Solve \(x + 7 > 3\).
Quick check+10 XP
(Extended) Solve \(-2x < 8\).
Section 03
Double inequalities.
(Extended) An inequality with three parts, such as \(-3 \le 3x - 2 < 7\), is solved by doing the same thing to ALL three parts at once until \(x\) is alone in the middle.
Worked example Three parts (Extended)
Solve \(-3 \le 3x - 2 < 7\) and list the integer solutions.
Add 2 to all three parts: \(-1 \le 3x < 9\).
Divide all three parts by 3: \(-\dfrac{1}{3} \le x < 3\).
Integers from \(-\dfrac{1}{3}\) up to, but not including, 3: \(0, 1, 2\).
Answer: \(-\dfrac{1}{3} \le x < 3\); integers \(0, 1, 2\)
Worked example With a fraction (Extended)
Solve \(1 < \dfrac{x + 5}{2} \le 4\).
Multiply all three parts by 2: \(2 < x + 5 \le 8\).
Subtract 5 from all three parts: \(-3 < x \le 3\).
Answer: \(-3 < x \le 3\)
Your turn
(Extended) Find the largest integer \(n\) such that \(7n - 3 < 40\).
\(7n < 43\), so \(n < \dfrac{43}{7} = 6.14...\). The largest integer is \(n = 6\).
Examiner trap
Do not split a double inequality and then only finish one half. Whatever you do to the middle, do to both ends as well, or the answer range is wrong.
Section 04
Writing your own inequality.
(Extended) Words such as 'at most', 'no more than' and 'cannot exceed' mean \(\le\); 'at least' and 'no fewer than' mean \(\ge\). Write the inequality, solve it, then answer the question in context.
Worked example Shopping budget (Extended)
Ali has $50. He buys a book for $8 and some pens at $3.20 each. Find the greatest number of pens he can buy.
Let \(p\) be the number of pens. Total cost is at most $50: \(8 + 3.2p \le 50\).
Subtract 8: \(3.2p \le 42\).
Divide by 3.2: \(p \le 13.125\).
\(p\) must be a whole number, so round DOWN: 14 pens would cost too much.
Answer: 13 pens
Examiner trap
In context the answer is often a whole number. Do not round 13.125 to the nearest integer by habit: think about which way keeps the inequality true. For 'at most' round down; for 'at least' round up.
Stage 2 · Exam
Exam-style questions
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(Extended) Solve \(7 - 3x \le 2x - 8\).
Exam style+20 XP
(Extended) List all the integers that satisfy \(-5 < 2x + 1 \le 7\).
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(Extended) Find the smallest integer \(x\) such that \(5(x - 2) > 2x + 3\).
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(Extended) A taxi charges $3 plus $1.80 per kilometre. Sara has $20. What is the greatest whole number of kilometres she can travel?
Skill unlocked
Inequalities, solved and drawn.
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