Quadratic Inequalities

What you'll learn

How to solve quadratic inequalities by finding their boundary values.
How to decide whether the answer is between or outside the roots.
How to handle strict and inclusive inequality signs.
How to list integer solutions when the answer is a range.

1. Start with the related quadratic equation

A quadratic inequality looks like a quadratic expression with an inequality sign instead of an equals sign.

Definition

Quadratic inequality

A quadratic inequality is an inequality involving a squared variable, such as $x^2+4x+3>0$ or $2x^2-11x+9<0$ .

To solve one, first find the boundary values. These are the values of $x$ where the quadratic expression equals zero. They split the number line into regions.

A root is a value of $x$ that makes an expression equal zero.

Example

Finding boundary values

Find the boundary values for $x^2+7x+12>0$ .

The roots at x = -4 and x = -3 split the number line into three regions to test.

Temporarily replace the inequality sign with an equals sign:
$x^2+7x+12=0$
Factorise the quadratic:
$x^2+7x+12=(x+3)(x+4)$
Set each bracket equal to zero:
$\quad \text{or} \quad x+4=0$
The boundary values are $x = - 3$ and $x = - 4$ .

2. Decide where the quadratic is positive or negative

The graph of a quadratic is called a parabola. If the coefficient of $x^2$ is positive, the parabola opens upwards, like a smile.

That means:

the quadratic is positive outside the roots
the quadratic is negative between the roots

Key Idea

Inside or outside?

For an upward-opening quadratic:

An upward-opening parabola is above the x-axis outside its roots and below the x-axis between them. $> 0$ or $≥0\ge 0$ usually means outside the roots.
$< 0$ or $≤0\le 0$ usually means between the roots.

Example

A greater-than inequality

Solve $x^2+5x+6>0$ .

For this strict greater-than inequality, the solution is outside the roots -3 and -2 with open endpoints.

Find the boundary values by solving the related equation:
$x^2+5x+6=0$
Factorise:
$(x + 2) (x + 3) = 0$
The roots are $x = - 2$ and $x = - 3$ . Put them in order on the number line: -3, then -2.
Since the coefficient of $x^2$ is positive, the quadratic is positive outside the roots.
The solution is:
$\quad \text{or} \quad x>-2$

Common Mistake

Using ‘and’ instead of ‘or’

When the answer is outside the roots, you usually need or, not and. A value cannot be both less than -3 and greater than -2 at the same time.

3. Strict vs inclusive inequalities

The symbols matter:

$<$ and $>$ are strict: do not include the boundary values.
$≤\le$ and $≥\ge$ are inclusive: include the boundary values.

Example

An inclusive less-than inequality

Solve $x2−8x+15≤0x^2-8x+15\le 0$ .

For this inclusive less-than inequality, the solution is the closed interval from 3 to 5.

Solve the related equation:
$x^2-8x+15=0$
Factorise:
$(x - 3) (x - 5) = 0$
The boundary values are $x = 3$ and $x = 5$ .
The parabola opens upwards, so the quadratic is less than or equal to zero between the roots, including the roots.
The solution is:
$3≤x≤53\le x\le 5$

Tip

Open and closed endpoints

On a number line, use open circles for $<$ or $>$ , and closed circles for $≤\le$ or $≥\ge$ .

4. Special cases like $x2≤100x^2\le 100$

If there is no $x$ term, still treat it as a quadratic inequality. Do not forget the negative square root.

Example

A square-only inequality

Solve $x2≤64x^2\le 64$ .

The square-only inequality has two boundary values, -8 and 8, not just the positive one.

Move everything to one side:
$x2−64≤0x^2-64\le 0$
Factorise using difference of two squares:
$(x−8)(x+8)≤0(x-8)(x+8)\le 0$
The boundary values are $x = - 8$ and $x = 8$ .
The parabola opens upwards, so the expression is less than or equal to zero between the roots.
The solution is:
$−8≤x≤8-8\le x\le 8$

Common Mistake

Forgetting the negative root

$x2≤64x^2\le 64$ does not just mean $x≤8x\le 8$ . For example, $x = - 5$ also works because squaring it gives a positive value.

5. Rearrange first if needed

Sometimes the inequality is not already equal to zero on one side. Rearrange it first, just like you would with an equation.

Example

Variables on both sides

Solve $5x+14>x^2$ .

After rearranging, the upward-opening quadratic is negative between the roots -2 and 7.

Bring all terms to one side so the other side is zero:
$x^2-5x-14<0$
Factorise:
$(x - 7) (x + 2) < 0$
The boundary values are $x = 7$ and $x = - 2$ .
The coefficient of $x^2$ is positive, so the expression is negative between the roots.
The solution is:
$- 2 < x < 7$

Common Mistake

Multiplying by -1

If you multiply or divide an inequality by a negative number, the inequality sign reverses. For example, $- x > 4$ becomes $x < - 4$ .

6. Non-monic quadratic inequalities

A non-monic quadratic has a coefficient of $x^2$ that is not one, such as $6x^2+7x-3$ .

The method is the same: factorise, find the boundary values, then choose the correct region.

Example

A non-monic inequality

Solve $6x^2+7x-3<0$ .

$The non-monic quadratic is below zero between the fractional roots -3/2 and 1/3.$

Factorise the quadratic:
$6x^2+7x-3=(3x-1)(2x+3)$
Set each bracket equal to zero to find the boundary values:
$\quad \text{or} \quad 2x+3=0$
Solve each equation:
$x=13orx=−32x=\frac{1}{3} \quad \text{or} \quad x=-\frac{3}{2}$
The parabola opens upwards, so the expression is negative between the roots.
The solution is:
$−32<x<13-\frac{3}{2}<x<\frac{1}{3}$

7. Integer solutions

An integer is a whole number, including negative numbers and zero. If a question asks for integer values, first solve the inequality as a range, then list the integers inside that range.

Sometimes the quadratic does not factorise nicely, so you may need the quadratic formula.

Example

Listing integer solutions

Find the integer values that satisfy $x^2-6x+7<0$ .

Only the integers lying strictly between the approximate roots 1.59 and 4.41 satisfy the inequality.

Find the boundary values using the quadratic formula:
$x=6±62−4⋅1⋅72⋅1x=\frac{6\pm\sqrt{6^2-4\cdot 1\cdot 7}}{2\cdot 1}$
Simplify the roots:
$x=3±2x=3\pm\sqrt{2}$
Because the parabola opens upwards and the inequality is less than zero, the solution lies between the roots:
$3−2<x<3+23-\sqrt{2}<x<3+\sqrt{2}$
Approximate the roots: $3−2≈1.593-\sqrt{2}\approx 1.59$ and $3+2≈4.413+\sqrt{2}\approx 4.41$ .
The integer values are 2, 3 and 4.

Exam technique

In the exam

Rearrange so the inequality has zero on one side.
Find the boundary values by factorising or using the quadratic formula.
Decide whether the answer is between the roots or outside the roots.
Check whether the inequality sign is strict or inclusive.
If asked for integer values, list only the whole numbers in the solution range.

Self review

Check yourself

If an upward-opening quadratic has roots at $x = 2$ and $x = 9$ , where is it greater than zero?
What is the difference between solving $x^2-4x+3<0$ and $x2−4x+3≤0x^2-4x+3\le 0$ ?
Why must you rearrange $8x+20>x^2$ before factorising?

Quadratic Inequalities

What you'll learn

How to solve quadratic inequalities by finding their boundary values.
How to decide whether the answer is between or outside the roots.
How to handle strict and inclusive inequality signs.
How to list integer solutions when the answer is a range.

1. Start with the related quadratic equation

A quadratic inequality looks like a quadratic expression with an inequality sign instead of an equals sign.

Definition

Quadratic inequality

A quadratic inequality is an inequality involving a squared variable, such as $x^2+4x+3>0$ or $2x^2-11x+9<0$ .

To solve one, first find the boundary values. These are the values of $x$ where the quadratic expression equals zero. They split the number line into regions.

A root is a value of $x$ that makes an expression equal zero.

Example

Finding boundary values

Find the boundary values for $x^2+7x+12>0$ .

The roots at x = -4 and x = -3 split the number line into three regions to test.

Temporarily replace the inequality sign with an equals sign:
$x^2+7x+12=0$
Factorise the quadratic:
$x^2+7x+12=(x+3)(x+4)$
Set each bracket equal to zero:
$\quad \text{or} \quad x+4=0$
The boundary values are $x = - 3$ and $x = - 4$ .

2. Decide where the quadratic is positive or negative

The graph of a quadratic is called a parabola. If the coefficient of $x^2$ is positive, the parabola opens upwards, like a smile.

That means:

the quadratic is positive outside the roots
the quadratic is negative between the roots

Key Idea

Inside or outside?

For an upward-opening quadratic:

Example

A greater-than inequality

Solve $x^2+5x+6>0$ .

For this strict greater-than inequality, the solution is outside the roots -3 and -2 with open endpoints.

Find the boundary values by solving the related equation:
$x^2+5x+6=0$
Factorise:
$(x + 2) (x + 3) = 0$
The roots are $x = - 2$ and $x = - 3$ . Put them in order on the number line: -3, then -2.
Since the coefficient of $x^2$ is positive, the quadratic is positive outside the roots.
The solution is:
$\quad \text{or} \quad x>-2$

Common Mistake

Using ‘and’ instead of ‘or’

When the answer is outside the roots, you usually need or, not and. A value cannot be both less than -3 and greater than -2 at the same time.

3. Strict vs inclusive inequalities

The symbols matter:

$<$ and $>$ are strict: do not include the boundary values.
$≤\le$ and $≥\ge$ are inclusive: include the boundary values.

Example

An inclusive less-than inequality

Solve $x2−8x+15≤0x^2-8x+15\le 0$ .

For this inclusive less-than inequality, the solution is the closed interval from 3 to 5.

Solve the related equation:
$x^2-8x+15=0$
Factorise:
$(x - 3) (x - 5) = 0$
The boundary values are $x = 3$ and $x = 5$ .
The parabola opens upwards, so the quadratic is less than or equal to zero between the roots, including the roots.
The solution is:
$3≤x≤53\le x\le 5$

Tip

Open and closed endpoints

On a number line, use open circles for $<$ or $>$ , and closed circles for $≤\le$ or $≥\ge$ .

4. Special cases like $x2≤100x^2\le 100$

If there is no $x$ term, still treat it as a quadratic inequality. Do not forget the negative square root.

Example

A square-only inequality

Solve $x2≤64x^2\le 64$ .

The square-only inequality has two boundary values, -8 and 8, not just the positive one.

Move everything to one side:
$x2−64≤0x^2-64\le 0$
Factorise using difference of two squares:
$(x−8)(x+8)≤0(x-8)(x+8)\le 0$
The boundary values are $x = - 8$ and $x = 8$ .
The parabola opens upwards, so the expression is less than or equal to zero between the roots.
The solution is:
$−8≤x≤8-8\le x\le 8$

Common Mistake

Forgetting the negative root

$x2≤64x^2\le 64$ does not just mean $x≤8x\le 8$ . For example, $x = - 5$ also works because squaring it gives a positive value.

5. Rearrange first if needed

Sometimes the inequality is not already equal to zero on one side. Rearrange it first, just like you would with an equation.

Example

Variables on both sides

Solve $5x+14>x^2$ .

After rearranging, the upward-opening quadratic is negative between the roots -2 and 7.

Bring all terms to one side so the other side is zero:
$x^2-5x-14<0$
Factorise:
$(x - 7) (x + 2) < 0$
The boundary values are $x = 7$ and $x = - 2$ .
The coefficient of $x^2$ is positive, so the expression is negative between the roots.
The solution is:
$- 2 < x < 7$

Common Mistake

Multiplying by -1

If you multiply or divide an inequality by a negative number, the inequality sign reverses. For example, $- x > 4$ becomes $x < - 4$ .

6. Non-monic quadratic inequalities

A non-monic quadratic has a coefficient of $x^2$ that is not one, such as $6x^2+7x-3$ .

The method is the same: factorise, find the boundary values, then choose the correct region.

Example

A non-monic inequality

Solve $6x^2+7x-3<0$ .

$The non-monic quadratic is below zero between the fractional roots -3/2 and 1/3.$

Factorise the quadratic:
$6x^2+7x-3=(3x-1)(2x+3)$
Set each bracket equal to zero to find the boundary values:
$\quad \text{or} \quad 2x+3=0$
Solve each equation:
$x=13orx=−32x=\frac{1}{3} \quad \text{or} \quad x=-\frac{3}{2}$
The parabola opens upwards, so the expression is negative between the roots.
The solution is:
$−32<x<13-\frac{3}{2}<x<\frac{1}{3}$

7. Integer solutions

An integer is a whole number, including negative numbers and zero. If a question asks for integer values, first solve the inequality as a range, then list the integers inside that range.

Sometimes the quadratic does not factorise nicely, so you may need the quadratic formula.

Example

Listing integer solutions

Find the integer values that satisfy $x^2-6x+7<0$ .

Only the integers lying strictly between the approximate roots 1.59 and 4.41 satisfy the inequality.

Find the boundary values using the quadratic formula:
$x=6±62−4⋅1⋅72⋅1x=\frac{6\pm\sqrt{6^2-4\cdot 1\cdot 7}}{2\cdot 1}$
Simplify the roots:
$x=3±2x=3\pm\sqrt{2}$
Because the parabola opens upwards and the inequality is less than zero, the solution lies between the roots:
$3−2<x<3+23-\sqrt{2}<x<3+\sqrt{2}$
Approximate the roots: $3−2≈1.593-\sqrt{2}\approx 1.59$ and $3+2≈4.413+\sqrt{2}\approx 4.41$ .
The integer values are 2, 3 and 4.

Exam technique

In the exam

Rearrange so the inequality has zero on one side.
Find the boundary values by factorising or using the quadratic formula.
Decide whether the answer is between the roots or outside the roots.
Check whether the inequality sign is strict or inclusive.
If asked for integer values, list only the whole numbers in the solution range.

Self review

Check yourself

If an upward-opening quadratic has roots at $x = 2$ and $x = 9$ , where is it greater than zero?
What is the difference between solving $x^2-4x+3<0$ and $x2−4x+3≤0x^2-4x+3\le 0$ ?
Why must you rearrange $8x+20>x^2$ before factorising?

Quadratic Inequalities

What you'll learn

1. Start with the related quadratic equation

2. Decide where the quadratic is positive or negative

3. Strict vs inclusive inequalities

4. Special cases like $x2≤100x^2\le 100$

5. Rearrange first if needed

6. Non-monic quadratic inequalities

7. Integer solutions

Recap questions

You've reached the end

Quadratic Inequalities

What you'll learn

1. Start with the related quadratic equation

2. Decide where the quadratic is positive or negative

3. Strict vs inclusive inequalities

4. Special cases like $x2≤100x^2\le 100$

5. Rearrange first if needed

6. Non-monic quadratic inequalities

7. Integer solutions

Recap questions

You've reached the end

Quadratic Inequalities

What you'll learn

1. Start with the related quadratic equation

2. Decide where the quadratic is positive or negative

3. Strict vs inclusive inequalities

4. Special cases like x2≤100x^2\le 100x2≤100

5. Rearrange first if needed

6. Non-monic quadratic inequalities

7. Integer solutions

Recap questions

You've reached the end

Quadratic Inequalities Revision Guide

Quadratic Inequalities

What you'll learn

1. Start with the related quadratic equation

2. Decide where the quadratic is positive or negative

3. Strict vs inclusive inequalities

4. Special cases like x2≤100x^2\le 100x2≤100

5. Rearrange first if needed

6. Non-monic quadratic inequalities

7. Integer solutions

Recap questions

You've reached the end

Quadratic Inequalities Revision Guide

4. Special cases like $x2≤100x^2\le 100$

4. Special cases like $x2≤100x^2\le 100$