Quadratic Inequalities

What is a Quadratic Inequality?

You can easily solve a standard equation like $x^2 = 4$ to find exactly where a curve crosses a line. But what happens when you need to know where the curve is greater or less than that line? Solving a quadratic inequality allows you to find an entire range of values that satisfy the condition, rather than just one or two specific points.

The first step is always to rearrange the inequality into the standard form $ax^2 + bx + c > 0$ (or $<, \leq, \geq$ ). It is standard practice to ensure the coefficient of $x^2$ is positive ( $a > 0$ ) to guarantee a "U-shaped" curve. If you need to multiply or divide the inequality by a negative number to achieve this, you must flip the direction of the inequality sign (for example, $-x^2 < -4$ becomes $x^2 > 4$ ).

Finding Critical Values

The critical values are the exact points where the quadratic expression equals zero. They mark the strict boundaries of your final solution set.

To find them, you temporarily replace the inequality sign with an equals sign to form the equation $ax^2 + bx + c = 0$ . You then solve this quadratic equation, usually by factorising into the form $(x \pm p)(x \pm q) = 0$ . If the quadratic cannot be factorised, you will need to use the quadratic formula or complete the square to find the critical values.

Visualising with a Parabola

Once you have your critical values, you must sketch a parabola. This sketch is a vital visual tool to determine which $x$ -values satisfy the inequality.

For $> 0$ or $\geq 0$ : The solution lies in the regions above the $x$ -axis. This forms the two outer "tails" of the parabola, creating disjoint intervals.
For $< 0$ or $\leq 0$ : The solution lies in the region below the $x$ -axis. This forms the single "valley" between the critical values, creating one continuous interval.

Disjoint intervals do NOT overlap and cannot mathematically be written as a single combined string (e.g., $5 < x < -2$ is forbidden). They must be separated by the word "or".

Representing the Solution Set

The solution set can be represented algebraically, on a number line, or on a graph using formal conventions:

Number Lines: Use open circles for strict inequalities ( $<, >$ ) and solid closed circles for inclusive inequalities ( $\leq, \geq$ ).
Graphs: Use dashed lines for strict inequalities and solid lines for inclusive ones. AQA examiner guidance advises shading the non-required (unwanted) region so the actual solution region remains clear.
Set Notation: This is a formal way of writing the solution set. It uses curly brackets { } and a colon : (meaning "such that"). Use the union symbol $\cup$ for "or" (disjoint regions) and the intersection symbol $\cap$ for "and".

Worked Example 1: Disjoint Regions ( $\geq 0$ )

Calculate the solution set for $x^2 - x - 12 \geq 0$ , giving your answer in set notation.

Step 1: Set to zero to find critical values.

$x^2 - x - 12 = 0$

Step 2: Factorise and solve.

$(x - 4)(x + 3) = 0$
The critical values are $x = 4$ and $x = -3$ .

Step 3: Sketch the parabola and identify the region.

Draw a positive (U-shaped) parabola crossing the $x$ -axis at $-3$ and $4$ .
Because the inequality is $\geq 0$ , look for the regions on or above the $x$ -axis.
These are the two outer "tails", which are disjoint.

Step 4: Express the final solution set.

Inequality notation: $x \leq -3$ or $x \geq 4$ .
Set notation: $\{x : x \leq -3\} \cup \{x : x \geq 4\}$ .

Worked Example 2: Single Interval ( $< 0$ )

Calculate the solution set for $x^2 - 3x - 10 < 0$ .

Step 1: Set to zero to find critical values.

$x^2 - 3x - 10 = 0$

Step 2: Factorise and solve.

$(x - 5)(x + 2) = 0$
The critical values are $x = 5$ and $x = -2$ .

Step 3: Sketch the parabola and identify the region.

Draw a positive parabola crossing the $x$ -axis at $-2$ and $5$ .
Because the inequality is $< 0$ , look for the region below the $x$ -axis.
This is the single "valley" between the two critical values.

Step 4: Express the final solution set.

Inequality notation: $-2 < x < 5$ .
Set notation: $\{x : -2 < x < 5\}$ .

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Exam Tips

Common Mistake
Students frequently use 'and' when they mean 'or' for disjoint regions (e.g., writing ' $x < -2$ and $x > 4$ '), which is mathematically impossible for a single value and will lose marks.
2
Always draw a quick sketch of the quadratic curve, as AQA examiners often award method marks (M1) for a correct sketch or identifying critical values, even if your final inequality signs are incorrect.
3
If you multiply or divide an inequality by a negative number to make the $x^2$ coefficient positive, you must remember to flip the direction of the inequality sign.
4
For AQA Higher tier questions asking specifically for 'set notation', you must use formal syntax with curly brackets, such as $\{x : x < -3\} \cup \{x : x > 4\}$ , rather than just writing out the inequalities.

Key Terms(5)

Solution set: The complete range or discrete list of values that satisfy an inequality.
Critical values: The specific values of x where the quadratic expression equals zero, acting as the boundaries for the solution set.
Parabola: The U-shaped curve (when a > 0) representing the quadratic function; used to visualize which x-values satisfy the inequality.
Disjoint intervals: Two separate ranges of numbers that do not overlap, typical of "greater than" quadratic inequalities.
Set notation: A formal mathematical way of writing a solution set using curly brackets and logical symbols.

Previous NoteLinear Inequalities and Number Lines Next NoteGraphical Inequalities and Shaded Regions

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Key Terms(5)

Solution set: The complete range or discrete list of values that satisfy an inequality.
Critical values: The specific values of x where the quadratic expression equals zero, acting as the boundaries for the solution set.
Parabola: The U-shaped curve (when a > 0) representing the quadratic function; used to visualize which x-values satisfy the inequality.
Disjoint intervals: Two separate ranges of numbers that do not overlap, typical of "greater than" quadratic inequalities.
Set notation: A formal mathematical way of writing a solution set using curly brackets and logical symbols.

Quadratic Inequalities

What is a Quadratic Inequality?

Finding Critical Values

The critical values are the exact points where the quadratic expression equals zero. They mark the strict boundaries of your final solution set.

Visualising with a Parabola

Once you have your critical values, you must sketch a parabola. This sketch is a vital visual tool to determine which $x$ -values satisfy the inequality.

For $> 0$ or $\geq 0$ : The solution lies in the regions above the $x$ -axis. This forms the two outer "tails" of the parabola, creating disjoint intervals.
For $< 0$ or $\leq 0$ : The solution lies in the region below the $x$ -axis. This forms the single "valley" between the critical values, creating one continuous interval.

Disjoint intervals do NOT overlap and cannot mathematically be written as a single combined string (e.g., $5 < x < -2$ is forbidden). They must be separated by the word "or".

Representing the Solution Set

The solution set can be represented algebraically, on a number line, or on a graph using formal conventions:

Number Lines: Use open circles for strict inequalities ( $<, >$ ) and solid closed circles for inclusive inequalities ( $\leq, \geq$ ).
Graphs: Use dashed lines for strict inequalities and solid lines for inclusive ones. AQA examiner guidance advises shading the non-required (unwanted) region so the actual solution region remains clear.
Set Notation: This is a formal way of writing the solution set. It uses curly brackets { } and a colon : (meaning "such that"). Use the union symbol $\cup$ for "or" (disjoint regions) and the intersection symbol $\cap$ for "and".

Worked Example 1: Disjoint Regions ( $\geq 0$ )

Calculate the solution set for $x^2 - x - 12 \geq 0$ , giving your answer in set notation.

Step 1: Set to zero to find critical values.

$x^2 - x - 12 = 0$

Step 2: Factorise and solve.

$(x - 4)(x + 3) = 0$
The critical values are $x = 4$ and $x = -3$ .

Step 3: Sketch the parabola and identify the region.

Draw a positive (U-shaped) parabola crossing the $x$ -axis at $-3$ and $4$ .
Because the inequality is $\geq 0$ , look for the regions on or above the $x$ -axis.
These are the two outer "tails", which are disjoint.

Step 4: Express the final solution set.

Inequality notation: $x \leq -3$ or $x \geq 4$ .
Set notation: $\{x : x \leq -3\} \cup \{x : x \geq 4\}$ .

Worked Example 2: Single Interval ( $< 0$ )

Calculate the solution set for $x^2 - 3x - 10 < 0$ .

Step 1: Set to zero to find critical values.

$x^2 - 3x - 10 = 0$

Step 2: Factorise and solve.

$(x - 5)(x + 2) = 0$
The critical values are $x = 5$ and $x = -2$ .

Step 3: Sketch the parabola and identify the region.

Draw a positive parabola crossing the $x$ -axis at $-2$ and $5$ .
Because the inequality is $< 0$ , look for the region below the $x$ -axis.
This is the single "valley" between the two critical values.

Step 4: Express the final solution set.

Inequality notation: $-2 < x < 5$ .
Set notation: $\{x : -2 < x < 5\}$ .

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students frequently use 'and' when they mean 'or' for disjoint regions (e.g., writing ' $x < -2$ and $x > 4$ '), which is mathematically impossible for a single value and will lose marks.
2
Always draw a quick sketch of the quadratic curve, as AQA examiners often award method marks (M1) for a correct sketch or identifying critical values, even if your final inequality signs are incorrect.
3
If you multiply or divide an inequality by a negative number to make the $x^2$ coefficient positive, you must remember to flip the direction of the inequality sign.
4
For AQA Higher tier questions asking specifically for 'set notation', you must use formal syntax with curly brackets, such as $\{x : x < -3\} \cup \{x : x > 4\}$ , rather than just writing out the inequalities.

Key Terms(5)

Solution set: The complete range or discrete list of values that satisfy an inequality.
Critical values: The specific values of x where the quadratic expression equals zero, acting as the boundaries for the solution set.
Parabola: The U-shaped curve (when a > 0) representing the quadratic function; used to visualize which x-values satisfy the inequality.
Disjoint intervals: Two separate ranges of numbers that do not overlap, typical of "greater than" quadratic inequalities.
Set notation: A formal mathematical way of writing a solution set using curly brackets and logical symbols.

Previous NoteLinear Inequalities and Number Lines Next NoteGraphical Inequalities and Shaded Regions

Back to Inequalities

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(5)

Solution set: The complete range or discrete list of values that satisfy an inequality.
Critical values: The specific values of x where the quadratic expression equals zero, acting as the boundaries for the solution set.
Parabola: The U-shaped curve (when a > 0) representing the quadratic function; used to visualize which x-values satisfy the inequality.
Disjoint intervals: Two separate ranges of numbers that do not overlap, typical of "greater than" quadratic inequalities.
Set notation: A formal mathematical way of writing a solution set using curly brackets and logical symbols.

Quadratic Inequalities

What is a Quadratic Inequality?

Finding Critical Values

Visualising with a Parabola

Representing the Solution Set

Worked Example 1: Disjoint Regions (≥0\geq 0≥0)

Worked Example 2: Single Interval (<0< 0<0)

Sign up to continue reading

Exam Tips

Key Terms(5)

Key Terms(5)

Quadratic Inequalities

What is a Quadratic Inequality?

Finding Critical Values

Visualising with a Parabola

Representing the Solution Set

Worked Example 1: Disjoint Regions (≥0\geq 0≥0)

Worked Example 2: Single Interval (<0< 0<0)

Sign up to continue reading

Exam Tips

Key Terms(5)

Key Terms(5)

Worked Example 1: Disjoint Regions ( $\geq 0$ )

Worked Example 2: Single Interval ( $< 0$ )

Worked Example 1: Disjoint Regions ( $\geq 0$ )

Worked Example 2: Single Interval ( $< 0$ )