Inequalities: Revision Notes

Condensed for the final weeks. For worked examples, use the Inequalities study guide.

The symbols

Symbol	Meaning	On a number line	On a graph
<	less than	Open circle ○	Dashed line
>	greater than	Open circle ○	Dashed line
≤	less than or equal	Filled circle ●	Solid line
≥	greater than or equal	Filled circle ●	Solid line

Open/dashed = the boundary is not included. Filled/solid = it is.

Solving linear inequalities

Solve exactly as you would an equation — with one critical exception:

Multiplying or dividing by a negative number REVERSES the inequality sign.

-3x > 12
  x < -4        <- sign flipped

This single rule accounts for most lost marks in the topic. Adding and subtracting never flip it.

Double inequalities

Operate on all three parts at once.

-5 <  2x + 1  <=  9
-6 <  2x      <=  8       (subtract 1 from each part)
-3 <  x       <=  4       (divide each part by 2)

Integer solutions

When asked to list integer values, apply the endpoints carefully.

For −3 < x ≤ 4, the integers are −2, −1, 0, 1, 2, 3, 4. Note −3 is excluded (strict) but 4 is included.

Regions on a graph

1. Draw each boundary line (dashed or solid as appropriate).
2. Decide which side satisfies the inequality — test a point, usually (0, 0) if it is not on the line.
3. Shade as the question instructs — read whether it asks you to shade the required region or the unwanted one.
4. Label the region R.

Test (0,0) in  y < 2x + 1:
   0 < 1  TRUE  ->  the origin side is the required region

Common region boundaries: x ≥ 0, y ≥ 0, x + y ≤ 10, y ≤ 2x.

Worked example. Represent x < 1 and y ≥ 1 on the same diagram.

Draw a broken vertical line at x = 1 (strict, so broken), and shade the unwanted side (x ≥ 1, to the right). Draw a solid horizontal line at y = 1 (inclusive, so solid), and shade the unwanted side (y < 1, below it). What’s left unshaded — to the left of x = 1 and on or above y = 1 — is the region satisfying both inequalities.

The reverse skill: reading inequalities from a region

Given a shaded diagram, identify each boundary line’s equation, decide whether it should be < / > (broken line) or ≤ / ≥ (solid line), and decide which side of each line the unshaded (wanted) region lies on — that determines the direction of each inequality sign.

Note: linear programming — optimising a quantity subject to a system of inequalities — is explicitly not part of this syllabus; the skill required stops at representing, solving and reading off regions.

Inequalities in worded problems

Translate the words into symbols before solving, and define what the variable represents.

Worked example. A taxi charges a $3 call-out fee plus $2 per kilometre. A passenger has at most $25 to spend. Form and solve an inequality for the number of kilometres, k, the passenger can travel.

3 + 2k <= 25
    2k <= 22
     k <= 11

The passenger can travel at most 11 km. Note the answer is bounded by the real-world context — k also cannot be negative, so the full solution is 0 <= k <= 11, even though the algebra alone only produces the upper bound.

Combining two separate inequalities

Some questions give two inequalities in the same variable and ask for the values that satisfy both.

Solve:  2x - 1 > 5   and   x + 4 <= 10

2x - 1 > 5            x + 4 <= 10
    2x > 6                 x <= 6
     x > 3

Combined: 3 < x <= 6

Write the combined answer as a single double inequality, not as two separate lines — this is what “hence find the set of values” is asking for.

Exam traps

• Forgetting to flip the sign when dividing by a negative.
• Using an open circle where ≤ requires a filled one.
• Dashed vs solid lines — worth a mark on its own.
• Shading the wrong region: always test a point rather than guessing.
• When listing integers, checking whether each endpoint is included.
• Treating a double inequality one side at a time and losing a bound in the process.

Checking a solution

After solving, substitute a value from your solution set back into the original inequality to confirm it holds — this catches sign errors quickly, especially after dividing by a negative number.

Check x < -4 solves -3x > 12:
   Try x = -5:  -3(-5) = 15,  and 15 > 12  TRUE

If the check fails, the sign was very likely not flipped when it should have been (or was flipped when it shouldn’t have been).

Self-test

1. Solve 4x − 3 ≤ 17.
2. Solve −2x > 10.
3. List the integers satisfying −2 ≤ x < 3.
4. Should y > x + 1 be drawn with a dashed or solid line?
5. How do you decide which side of a line to shade?

Answers: 1. 4x ≤ 20 → x ≤ 5. 2. Divide by −2 and flip: x < −5. 3. −2, −1, 0, 1, 2. 4. Dashed — the inequality is strict, so points on the line are not included. 5. Substitute a test point not on the line (usually the origin) into the inequality; if it is true, shade that side.

For the full worked explanation with additional detail, see the Inequalities study guide; for exam-style questions with full mark schemes, see the Inequalities practice questions.