IGCSE Maths inequalities and regions: solve, show on a number line, shade

What each tier expects

The key ideas

An inequality describes a range of values rather than one value. x > 3 means every number bigger than 3, not including 3. x ≤ 3 includes 3. A double inequality such as -1 ≤ x < 3 means both conditions hold, so x is between -1 and 3, including -1 but not 3.

You may add or subtract the same number on both sides, and multiply or divide by a positive number, without changing the sign. Multiplying or dividing by a negative flips the order of numbers on the number line, so the sign must reverse: from -x > 2 you get x < -2.

On a coordinate grid, the line y = 2x + 1 splits the plane in two. Every point above it satisfies y > 2x + 1 and every point below satisfies y < 2x + 1. To decide which side you want, test a point that is not on the line, usually (0, 0). Several inequalities together define a region where all are true.

For a quadratic inequality (Edexcel Higher), find where the quadratic equals zero, sketch the U-shaped curve, and read off where it is above or below the x-axis. The answer is either one interval between the roots or two separate intervals outside them.

Worked example 1: solve -3 ≤ 3x - 2 < 7 and list the integer solutions

1. Add 2 to all three parts: -1 ≤ 3x < 9.
2. Divide all three parts by 3 (positive, so no reversal): -1/3 ≤ x < 3.
3. The integers in this range are 0, 1 and 2. Note that -1/3 is not an integer and 3 is excluded.
4. Number line: closed circle at -1/3, open circle at 3, line joining them.
5. Check x = 2: 3(2) - 2 = 4, and -3 ≤ 4 < 7 is true.

Worked example 2: solve 7 - x ≥ 5

1. Subtract 7 from both sides: -x ≥ -2.
2. Divide both sides by -1 and reverse the sign: x ≤ 2.
3. Check with x = 0, which should work: 7 - 0 = 7 ≥ 5. True. Check with x = 3, which should fail: 7 - 3 = 4, which is not ≥ 5. Correct.
4. A safer route that avoids dividing by a negative: add x to both sides to get 7 ≥ 5 + x, then subtract 5 to get 2 ≥ x, the same answer.

Worked example 3 (Edexcel Higher): solve x^2 + 3x + 2 > 0

1. Solve the equation x^2 + 3x + 2 = 0: (x + 1)(x + 2) = 0, so x = -1 or x = -2.
2. Sketch y = (x + 1)(x + 2): a U-shaped curve crossing the x-axis at -2 and -1.
3. We want y > 0, which is where the curve is above the axis: to the left of -2 and to the right of -1.
4. Answer: x < -2 or x > -1. Check x = 0: 0 + 0 + 2 = 2 > 0, true. Check x = -1.5 (between the roots): 2.25 - 4.5 + 2 = -0.25, not > 0, so the middle is correctly excluded.

Worked example 4: a region on a grid

1. Show the region defined by x ≥ 0, y ≥ 1 and x + y ≤ 5.
2. Draw the boundaries: the y-axis (x = 0), the horizontal line y = 1, and the line x + y = 5 through (0, 5) and (5, 0). All three are inclusive, so in Cambridge style they are solid lines.
3. Test (0, 0) in x + y ≤ 5: 0 ≤ 5 is true, so the wanted side of that line is towards the origin. For y ≥ 1, the wanted side is above y = 1.
4. The region is the triangle with vertices (0, 1), (0, 5) and (4, 1). Cambridge candidates shade everything outside it unless told to shade the region itself; always follow the instruction in the question.

Common mistakes that cost marks

• Forgetting to reverse the sign after dividing by a negative.
• Writing the answer as an equation, x = 2, instead of an inequality.
• Using the wrong circle on a number line: open for < and >, closed for ≤ and ≥.
• Shading the wrong side of a line because no test point was used.
• For quadratic inequalities, writing -2 < x > -1 or combining two separate intervals into one.
• Listing integers that are excluded, such as including 3 when the inequality says x < 3.

Exam technique and how a tutor helps

Read whether the question wants the solution set, the integer values, or a region, because they are marked differently. For regions on a Cambridge paper, check the instruction: by default the unwanted region is shaded and the wanted one left clear. Label the region R if asked. When the question gives a diagram and asks for the inequalities that define a shaded region, find each boundary line's equation first, then use a test point to choose the sign.

Questions on this topic are short but easy to drop marks on through a single sign. Writing a quick check of one value inside and one outside your answer takes seconds.

In one-to-one lessons a tutor can show the idea behind the sign reversal on a number line rather than asking the student to memorise it, then use the shared whiteboard to draw regions together, line by line. That visual step is where most students go wrong, and it is much easier to correct live than from marked homework.

Self-check: can you do these?

• Solve 3x < 2x + 4. (Answer: x < 4)
• Solve 3 < x + 2 ≤ 5. (Answer: 1 < x ≤ 3)
• Solve 4x^2 > 25. (Answer: x < -2.5 or x > 2.5)
• Solve x^2 ≤ 25. (Answer: -5 ≤ x ≤ 5)
• Write down the three inequalities that define the triangle with vertices (0, 0), (4, 0) and (0, 4). (Answer: x ≥ 0, y ≥ 0, x + y ≤ 4)