Solving an inequality means finding every value of x that works and writing the answer as an interval or a union of intervals. At A-Level you need linear and quadratic inequalities, including those with a negative x² coefficient, inequalities comparing a curve with a line, reciprocal inequalities such as 4/x² > 1, and sets of inequalities that describe a shaded region.
Key rules for inequalities
- Multiplying or dividing by a negative number reverses the inequality
- Quadratic: find the roots, sketch the curve, read off the x-values
- For a U-shaped curve: < 0 between the roots, > 0 outside the roots
- f(x) < g(x) where the graph of f lies below the graph of g
Answers outside the roots are two intervals, such as x < −2 or x > 3, which can also be written as {x : x < −2} ∪ {x : x > 3}.
Linear inequalities
Solve a linear inequality like an equation, with one extra rule: if you multiply or divide both sides by a negative number, the inequality sign reverses. You can often avoid this by collecting the x terms on the side where their coefficient stays positive.
Quadratic inequalities
Rearrange so one side is zero, find the roots of the quadratic, and sketch the curve. For a U-shaped parabola, the curve is below zero between the roots and above zero outside them. So x² − x − 6 < 0 gives −2 < x < 3, and x² − x − 6 > 0 gives x < −2 or x > 3.
If the x² coefficient is negative, multiply through by −1 first, reversing the sign, so you are always reading a U-shaped curve. A sketch takes seconds and prevents the most common error: choosing the inside when you needed the outside.
A curve against a line or another curve
To solve f(x) < g(x), find where the graphs meet; those x-values are the roots of f(x) − g(x) = 0 and form the boundaries of the interval. Then decide where the graph of f lies below the graph of g. Bringing everything to one side turns it into an ordinary quadratic inequality.
Reciprocal inequalities
For an inequality such as 4/x² > 1, notice that x² is positive for every x except 0, so you can multiply through by x² without reversing the sign. Solve the resulting quadratic inequality, then exclude x = 0, where the original expression is undefined. If the denominator could be negative, such as with 1/x, multiply by x² rather than x to keep the sign safe.
Describing a shaded region
A region R bounded by curves and lines is described by one inequality for each boundary. Above a graph means y > or y ≥ that expression; below means y < or y ≤. Use ≥ or ≤ if the boundary is drawn solid and included, and > or < if it is dashed. Test a point inside R in each inequality to check the directions.
Worked examples
Example 1. Solve 4x − 3 ≥ 6x + 7.
- Collect x terms on the right to keep them positive: −3 − 7 ≥ 6x − 4x.
- −10 ≥ 2x.
- Divide by 2: −5 ≥ x.
Answer: x ≤ −5.
Marks: M1 for collecting terms, A1 for x ≤ −5.
Example 2. Solve x² − x − 6 > 0.
- Factorise: (x − 3)(x + 2) > 0, so the roots are 3 and −2.
- The curve is U-shaped, so it is above zero outside the roots.
Answer: x < −2 or x > 3.
Marks: M1 for the roots, A1 for the outside region.
Example 3. Find the values of x for which x² − 2x − 3 < x − 3.
- Bring everything to one side: x² − 3x < 0.
- Factorise: x(x − 3) < 0, so the roots are 0 and 3.
- Between the roots, the expression is negative.
Answer: 0 < x < 3.
Marks: M1 for one side equal to 0, A1 for 0 < x < 3.
Example 4. Solve 4/x² > 1.
- x² > 0 for x ≠ 0, so multiply through without reversing: 4 > x².
- x² < 4 gives −2 < x < 2.
- Exclude x = 0, where 4/x² is undefined.
Answer: −2 < x < 2, with x ≠ 0.
Marks: M1 for multiplying by x², A1 for the interval with the exclusion.
Example 5. The region R lies above the curve y = x² − 2x − 2 and below the line y = x + 2, with both boundaries included. Write the inequalities that define R.
- R is above the curve, so y ≥ x² − 2x − 2.
- R is below the line, so y ≤ x + 2.
Answer: y ≥ x² − 2x − 2 and y ≤ x + 2.
Marks: B1 for each inequality.
Common mistakes
- Not reversing the sign. Multiplying or dividing by a negative reverses the inequality.
- Choosing the wrong region. Sketch the curve: a U-shaped curve is negative between its roots.
- Writing an impossible combined inequality. x < −2 or x > 3 cannot be written as 3 < x < −2; use “or” or set notation.
- Multiplying by x when x could be negative. Multiply by x² instead, which is always positive for x ≠ 0.
- Forgetting the excluded value. Values that make a denominator zero must be excluded.
How inequalities are examined at A-Level
Inequalities form part of the equations and inequalities chapter in Pure 1 (WMA11) for Edexcel International A-Level, and of first-year Pure for Edexcel A-Level (9MA0). Questions ask you to:
- solve linear and quadratic inequalities, sometimes together
- find where a curve lies below a line, after finding the intersection points
- solve a reciprocal inequality, stating any excluded value
- write the inequalities that define a shaded region
The Pure 1 equations and inequalities exam questions include all of these, with mark schemes. For extra practice use the equations and inequalities worksheet or the algebra revision engine.
Quadratic inequalities are essential for the discriminant and for increasing and decreasing functions. For the GCSE starting point, see quadratic inequalities.
Frequently asked questions
How do you solve a quadratic inequality?
Rearrange so one side is zero, find the roots, sketch the curve, and read off where it is above or below the x-axis.
How do you write an answer with two intervals?
Use “or”, such as x < −2 or x > 3, or set notation: {x : x < −2} ∪ {x : x > 3}.
Why can you multiply by x² but not by x?
x² is always positive (for x ≠ 0), so the inequality keeps its direction. x could be negative, which would reverse it.