Edexcel International GCSE Mathematics A 4MA1: Inequalities – Revision Notes

For full explanations and worked examples, use the inequalities study guide. These notes condense section 2.8 Inequalities of Topic 2 in the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series examined on it. Foundation statements 2.8A–E are needed on both tiers. Higher statements 2.8A (quadratic inequalities) and 2.8B (harder regions) are Higher tier only. A calculator may be used on every paper.

Course hub: Edexcel IGCSE Mathematics. Tick off each statement on the printable checklist, then test yourself with the inequalities practice questions or a free diagnostic. For factorising and completing the square, which the Higher work relies on, see the algebraic manipulation revision notes.

Key terms

• Inequality: a statement that one quantity is bigger or smaller than another, using >, <, ≥ or ≤.
• Strict inequality: > or <. The boundary value is not included.
• Inclusive inequality: ≥ or ≤. The boundary value is included.
• Double-ended inequality: a lower and an upper limit in one statement, e.g. −1 ≤ x < 7.
• Solution set: every value that makes the inequality true. Usually a range, not one number.
• Critical values (Higher tier only): the roots of the related quadratic equation. They split the number line into the parts you test.
• Region: the part of a graph where every point satisfies a set of inequalities in x and y.
• Boundary line: the line you get by replacing the inequality sign with =.

Symbols and number lines (2.8A, 2.8B, both tiers)

Inequality	Words	Number line
x > a	greater than a	open circle at a, arrow right
x ≥ a	greater than or equal to a	closed circle at a, arrow right
x < a	less than a	open circle at a, arrow left
x ≤ a	less than or equal to a	closed circle at a, arrow left
a < x ≤ b	between a and b, including b	open at a, closed at b, joined

Open circle = not included. Closed circle = included.

When you list integers, check each end on its own. For −1 < n ≤ 3 the integers are 0, 1, 2, 3: −1 is out, 3 is in.

Method in steps

Linear inequality (2.8C, both tiers)

1. Expand brackets and clear fractions (multiply every term by the denominator).
2. Collect x-terms on one side and numbers on the other.
3. Divide by the coefficient of x. If it is negative, reverse the sign.
4. Write the answer as x > …, x ≤ …, and so on.
5. Show it on a number line if asked.

Double-ended inequality (2.8C, both tiers)

1. Do the same operation to all three parts.
2. Aim to leave x alone in the middle.
3. If you divide all three parts by a negative, reverse both signs and rewrite in ascending order: −6 < −2x ≤ 4 gives 3 > x ≥ −2, which you write as −2 ≤ x < 3.

Shading a region (2.8D, both tiers)

1. Replace each inequality sign with = and draw each boundary line.
2. For each line, substitute a test point not on the line (use (0, 0) if you can).
3. If the test point makes the inequality true, the region is on its side; otherwise it is on the other side.
4. The required region is where all the conditions overlap. Shade or label it as the question instructs.

The specification says conventions for showing whether a boundary is included are not required.

Finding the inequalities for a region (2.8E, both tiers)

1. Write the equation of each boundary line.
2. Choose one point clearly inside the region.
3. Substitute it into each line’s expression to pick the sign.

Quadratic inequality (Higher tier only)

1. Rearrange to ax² + bx + c (sign) 0.
2. Solve ax² + bx + c = 0 for the critical values (factorise, formula or completing the square).
3. Sketch the U-shaped graph (for a > 0) through the critical values.
4. < 0 or ≤ 0 → between the roots. > 0 or ≥ 0 → outside the roots.
5. Write “between” as one double inequality; write “outside” as two inequalities joined by or.

Must-know distinctions

	This	Not this
Sign reversal	Multiplying or dividing by a negative	Adding or subtracting a negative
x² < 16	−4 < x < 4	x < 4 or x < ±4
x² > 16	x < −4 or x > 4	x > ±4 or −4 > x > 4
“Between” answer	p < x < q	x < q only (half the answer)
“Outside” answer	x < p or x > q	p > x > q (impossible)
y ≥ 2x + 1	above the line	below the line
x ≤ 3 on a graph	left of the vertical line x = 3	below the horizontal line y = 3

Small worked reminders

Linear with brackets (both tiers). Solve 3(x − 2) < 5x + 4.

3x − 6 < 5x + 4
   −10 < 2x
    −5 < x        so  x > −5

Keeping the x-term positive (moving 3x to the right) avoided dividing by a negative.

Double-ended with a negative coefficient (both tiers). Solve 1 < 7 − 2x ≤ 9.

 −6 < −2x ≤ 2      subtract 7 from all parts
  3 > x ≥ −1       divide by −2: both signs reverse
 −1 ≤ x < 3        rewrite in ascending order

Finding a vertex (both tiers). Where does y = 2x meet x + y = 9? Substitute: x + 2x = 9, so x = 3 and y = 6. The vertex is (3, 6).

Identifying a region (both tiers). A region is bounded by y = 2, x = −1 and y = −x + 6, and contains the point (1, 3). Test (1, 3) in each line:

y = 2:        3 is above 2               → y ≥ 2
x = −1:       1 is to the right of −1    → x ≥ −1
y = −x + 6:   3 compared with 5: smaller → y ≤ −x + 6

Harder region (Higher tier only). Lines written as ax + by = c are easiest to draw from their intercepts. For 4x + 5y = 20, put x = 0 to get (0, 4) and y = 0 to get (5, 0). Test (0, 0): 0 ≤ 20 is true, so 4x + 5y ≤ 20 is the side containing the origin. If a vertex is where two sloping lines cross, solve the two equations simultaneously rather than reading it from the grid.

Quadratic, “less than”, non-unit coefficient (Higher tier only). Solve 3x² − 7x + 2 < 0.

3x² − 7x + 2 = (3x − 1)(x − 2)
critical values: 1/3 and 2
"< 0" → between:  1/3 < x < 2

Quadratic with no factors (Higher tier only). For x² − 2x − 4 ≥ 0, completing the square gives (x − 1)² − 5, so the critical values are 1 ± √5. “≥ 0” → outside: x ≤ 1 − √5 or x ≥ 1 + √5, which is x ≤ −1.24 or x ≥ 3.24 to 3 significant figures.

Quick self-test

1. List the integers n such that −4 ≤ n < 1.
2. Solve 6x − 5 > 13.
3. Solve 9 − 2x ≤ 1.
4. Solve 1 ≤ 4x − 3 < 13.
5. A number line has a closed circle at −2 and an arrow pointing left. Write down the inequality it shows.
6. Does the point (2, 3) satisfy y < 2x − 1? Explain.
7. A triangular region has sides on x = 0, y = 0 and x + 2y = 6. Write down the three inequalities that define the region inside it.
8. (Higher) Solve x² > 64.
9. (Higher) Solve x² − 7x + 10 ≤ 0.
10. (Higher) Solve 2x² + x − 6 > 0.
11. (Higher) Solve x² < 5x.

Answers

1. −4, −3, −2, −1, 0
2. 6x > 18, so x > 3
3. −2x ≤ −8, so x ≥ 4 (sign reversed)
4. 4 ≤ 4x < 16, so 1 ≤ x < 4
5. x ≤ −2
6. 2(2) − 1 = 3 and 3 < 3 is false, so no. The point is on the boundary line, which a strict inequality excludes.
7. x ≥ 0, y ≥ 0, x + 2y ≤ 6 (vertices (0, 0), (6, 0) and (0, 3))
8. x < −8 or x > 8
9. (x − 2)(x − 5) ≤ 0, between the roots: 2 ≤ x ≤ 5
10. (2x − 3)(x + 2) > 0, outside the roots: x < −2 or x > 3/2
11. x² − 5x < 0, x(x − 5) < 0, between the roots: 0 < x < 5. Dividing both sides by x to get x < 5 is wrong, because x could be negative.

Where marks are usually lost

• Forgetting to reverse the sign after dividing by a negative, or reversing it after a subtraction.
• Giving a final answer with an = sign, such as x = 4, when the question asks you to solve an inequality.
• Using the wrong circle on a number line: filled for a strict inequality or open for an inclusive one.
• Operating on only two parts of a double-ended inequality.
• Listing integers that include an excluded end value, or missing 0 or negative integers.
• Shading the wrong side of a sloping line because no test point was used.
• Confusing x ≤ 3 (vertical line) with y ≤ 3 (horizontal line) when drawing regions.
• (Higher) Giving x² < 16 as x < 4, or writing x² > 16 as −4 > x > 4.
• (Higher) Not rearranging to “… 0” before factorising, for example setting each factor of x(x + 4) ≥ 12 against 12.
• (Higher) Dividing both sides of an inequality by x, which loses solutions when x could be negative.

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 2, Equations, formulae and identities, section 2.8 Inequalities (Foundation tier statements 2.8A–E and Higher tier statements 2.8A–B).