Inequalities

What you'll learn

What the inequality symbols mean, including strict and inclusive inequalities.
How to show inequalities on a number line using open and filled circles.
How to list integer solutions from an inequality.
How to solve linear inequalities, including double inequalities and brackets.

Inequalities are like equations with a range of answers

An equation usually has an equals sign and often one exact answer, like $x = 4$ .

An inequality compares values using signs such as $<$ , $>$ , $≤\le$ and $≥\ge$ . Instead of one answer, an inequality usually describes a range of possible answers.

Definition

Inequality symbols

$x < 3$ means $x$ is less than 3.
$x > 3$ means $x$ is greater than 3.
$x≤3x\le 3$ means $x$ is less than or equal to 3.
$x≥3x\ge 3$ means $x$ is greater than or equal to 3.

A solution to an inequality is any value that makes the inequality true.

Example

Checking whether values satisfy an inequality

For the inequality $x < 4$ , decide whether 2, 4 and 6 are solutions.

Substitute 2 into the inequality.

Since 2 is less than 4, 2 is a solution.
Substitute 4 into the inequality.

Since $x < 4$ does not include 4 itself, 4 is not a solution.
Substitute 6 into the inequality.

Since 6 is greater than 4, 6 is not a solution.

Key Idea

Strict or inclusive?

The symbols $<$ and $>$ are strict, so the endpoint is not included. The symbols $≤\le$ and $≥\ge$ are inclusive, so the endpoint is included.

Showing inequalities on a number line

A number line is a line marked with numbers in order. It helps you show all the values that satisfy an inequality.

Use:

an open circle for $<$ or $>$ , because the endpoint is not included
a filled circle for $≤\le$ or $≥\ge$ , because the endpoint is included
an arrow when the values continue forever in one direction
a thick line segment when the values are between two endpoints

Number lines showing open and filled circles for inequalities

Example

Drawing $x > - 1$ on a number line

Find -1 on the number line.
Because the sign is $>$ , draw an open circle at -1.
Since $x$ is greater than -1, shade to the right.
Add an arrow to show the values continue to the right forever.

Example

Writing an inequality from a number line

A number line has a filled circle at -4 and an arrow going to the right.

The filled circle means -4 is included, so use $≥\ge$ rather than $>$ .
The arrow goes to the right, so the values are greater than or equal to -4.
Write the inequality as:
$x≥−4x\ge -4$

Common Mistake

Open circle vs filled circle

Do not use a filled circle for $x < 2$ or $x > 2$ . A filled circle would include 2, but strict inequalities do not include the endpoint.

Listing integer values

An integer is a whole number, including negative whole numbers and zero. Examples include -3, -2, -1, 0, 1, 2 and 3.

When a question says the variable is an integer, you only list whole-number values that fit the inequality.

Example

Listing possible integer values

$n$ is an integer and $−3≤n<2-3\le n<2$ . List all possible values of $n$ .

Look at the left side: $−3≤n-3\le n$ means $n$ can be -3 or anything bigger.
Look at the right side: $n < 2$ means $n$ must be less than 2, so 2 is not included.
List the integers from -3 up to, but not including, 2.
The possible values are $n = - 3, - 2, - 1, 0, 1$ .

Tip

Check the endpoints first

Before listing integers, ask: “Is the left endpoint included?” and “Is the right endpoint included?” That usually prevents losing a mark.

Solving one-step and two-step inequalities

You solve many inequalities almost like equations: use inverse operations, which are operations that undo each other.

For example:

adding 7 is undone by subtracting 7
multiplying by 4 is undone by dividing by 4

Key Idea

Keep the inequality balanced

Whatever you do to one side of the inequality, do the same to the other side.

Example

Solving a two-step inequality

Solve $3t+5≤203t+5\le 20$ .

Subtract 5 from both sides.
$3t≤153t\le 15$
Divide both sides by 3.
$t≤5t\le 5$
The solution is all values of $t$ that are less than or equal to 5.

Example

Solving and showing on a number line

Solve $4x≤x+64x\le x+6$ and show the solution on a number line.

Subtract $x$ from both sides.
$3x≤63x\le 6$
Divide both sides by 3.
$x≤2x\le 2$
To show this on a number line, draw a filled circle at 2 and shade to the left.

Number line for the solution x less than or equal to 2

The sign changes when you divide by a negative

There is one very important difference between equations and inequalities.

Common Mistake

Dividing or multiplying by a negative

If you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.

For example, if you get $- 3 x < 12$ , dividing by -3 gives $x > - 4$ , not $x < - 4$ .

Example

Solving when the coefficient is negative

Solve $5 > 17 - 3 x$ .

Subtract 17 from both sides.
$- 12 > - 3 x$
Divide both sides by -3, remembering to reverse the inequality sign.
$4 < x$
Rewrite it in the more familiar order.
$x > 4$

Common Mistake

Forgetting to reverse the sign

The most common error is going from $- 12 > - 3 x$ to $4 > x$ . Because you divided by -3, the sign must reverse.

Double inequalities

A double inequality puts the variable between two bounds, such as $−2<x≤5-2<x\le 5$ .

This means both conditions must be true:

$x$ is greater than -2
$x$ is less than or equal to 5

Example

Drawing a double inequality

Show $−4<n≤3-4<n\le 3$ on a number line.

Mark -4 with an open circle because $n$ is greater than -4, not equal to -4.
Mark 3 with a filled circle because $n$ can equal 3.
Draw a thick line between -4 and 3.

Example

Listing integer values from a double inequality

$y$ is an integer and $2≤y<72\le y<7$ . List the possible values of $y$ .

The inequality starts at 2, and 2 is included.
The inequality ends before 7, so 7 is not included.
List the integers from 2 to 6.
The possible values are $y = 2, 3, 4, 5, 6$ .

Solving double inequalities

To solve a double inequality, do the same operation to all three parts: the left side, the middle, and the right side.

Example

Solving a double inequality for integers

$y$ is an integer and $3≤2y−1<113\le 2y-1<11$ . List all possible values of $y$ .

Add 1 to all three parts.
$4≤2y<124\le 2y<12$
Divide all three parts by 2.
$2≤y<62\le y<6$
Since $y$ is an integer, list the whole numbers from 2 up to, but not including, 6.
The possible values are $y = 2, 3, 4, 5$ .

Tip

Work on all three parts

For a double inequality, avoid only changing the middle expression. If you add 1 to the middle, you must add 1 to the left and the right as well.

Inequalities with brackets

Sometimes you must expand brackets before solving. Expanding means multiplying the term outside the bracket by every term inside the bracket.

Example

Solving an inequality with brackets

Solve $2 (3 n - 4) > 10$ .

Expand the bracket.
$6 n - 8 > 10$
Add 8 to both sides.
$6 n > 18$
Divide both sides by 6.
$n > 3$

Example

Solving when the variable is on both sides

Solve $5x−6≤2x+95x-6\le 2x+9$ .

Subtract $2 x$ from both sides.
$3x−6≤93x-6\le 9$
Add 6 to both sides.
$3x≤153x\le 15$
Divide both sides by 3.
$x≤5x\le 5$

Exam technique

In the exam

Decide whether each endpoint is included: open circle for $<$ or $>$ , filled circle for $≤\le$ or $≥\ge$ .
When solving, show each inverse operation clearly so the examiner can award method marks.
If you multiply or divide by a negative number, reverse the inequality sign immediately.

Self review

Check yourself

Can you explain the difference between $x < 3$ and $x≤3x\le 3$ on a number line?
If $n$ is an integer and $−2<n≤4-2<n\le 4$ , which values of $n$ are possible?
When solving $10 - 2 x < 4$ , why does the inequality sign change?

Inequalities

What you'll learn

What the inequality symbols mean, including strict and inclusive inequalities.
How to show inequalities on a number line using open and filled circles.
How to list integer solutions from an inequality.
How to solve linear inequalities, including double inequalities and brackets.

Inequalities are like equations with a range of answers

An equation usually has an equals sign and often one exact answer, like $x = 4$ .

An inequality compares values using signs such as $<$ , $>$ , $≤\le$ and $≥\ge$ . Instead of one answer, an inequality usually describes a range of possible answers.

Definition

Inequality symbols

$x < 3$ means $x$ is less than 3.
$x > 3$ means $x$ is greater than 3.
$x≤3x\le 3$ means $x$ is less than or equal to 3.
$x≥3x\ge 3$ means $x$ is greater than or equal to 3.

A solution to an inequality is any value that makes the inequality true.

Example

Checking whether values satisfy an inequality

For the inequality $x < 4$ , decide whether 2, 4 and 6 are solutions.

Substitute 2 into the inequality.

Since 2 is less than 4, 2 is a solution.
Substitute 4 into the inequality.

Since $x < 4$ does not include 4 itself, 4 is not a solution.
Substitute 6 into the inequality.

Since 6 is greater than 4, 6 is not a solution.

Key Idea

Strict or inclusive?

The symbols $<$ and $>$ are strict, so the endpoint is not included. The symbols $≤\le$ and $≥\ge$ are inclusive, so the endpoint is included.

Showing inequalities on a number line

A number line is a line marked with numbers in order. It helps you show all the values that satisfy an inequality.

Use:

an open circle for $<$ or $>$ , because the endpoint is not included
a filled circle for $≤\le$ or $≥\ge$ , because the endpoint is included
an arrow when the values continue forever in one direction
a thick line segment when the values are between two endpoints

Number lines showing open and filled circles for inequalities

Example

Drawing $x > - 1$ on a number line

Find -1 on the number line.
Because the sign is $>$ , draw an open circle at -1.
Since $x$ is greater than -1, shade to the right.
Add an arrow to show the values continue to the right forever.

Example

Writing an inequality from a number line

A number line has a filled circle at -4 and an arrow going to the right.

The filled circle means -4 is included, so use $≥\ge$ rather than $>$ .
The arrow goes to the right, so the values are greater than or equal to -4.
Write the inequality as:
$x≥−4x\ge -4$

Common Mistake

Open circle vs filled circle

Do not use a filled circle for $x < 2$ or $x > 2$ . A filled circle would include 2, but strict inequalities do not include the endpoint.

Listing integer values

An integer is a whole number, including negative whole numbers and zero. Examples include -3, -2, -1, 0, 1, 2 and 3.

When a question says the variable is an integer, you only list whole-number values that fit the inequality.

Example

Listing possible integer values

$n$ is an integer and $−3≤n<2-3\le n<2$ . List all possible values of $n$ .

Look at the left side: $−3≤n-3\le n$ means $n$ can be -3 or anything bigger.
Look at the right side: $n < 2$ means $n$ must be less than 2, so 2 is not included.
List the integers from -3 up to, but not including, 2.
The possible values are $n = - 3, - 2, - 1, 0, 1$ .

Tip

Check the endpoints first

Before listing integers, ask: “Is the left endpoint included?” and “Is the right endpoint included?” That usually prevents losing a mark.

Solving one-step and two-step inequalities

You solve many inequalities almost like equations: use inverse operations, which are operations that undo each other.

For example:

adding 7 is undone by subtracting 7
multiplying by 4 is undone by dividing by 4

Key Idea

Keep the inequality balanced

Whatever you do to one side of the inequality, do the same to the other side.

Example

Solving a two-step inequality

Solve $3t+5≤203t+5\le 20$ .

Subtract 5 from both sides.
$3t≤153t\le 15$
Divide both sides by 3.
$t≤5t\le 5$
The solution is all values of $t$ that are less than or equal to 5.

Example

Solving and showing on a number line

Solve $4x≤x+64x\le x+6$ and show the solution on a number line.

Subtract $x$ from both sides.
$3x≤63x\le 6$
Divide both sides by 3.
$x≤2x\le 2$
To show this on a number line, draw a filled circle at 2 and shade to the left.

Number line for the solution x less than or equal to 2

The sign changes when you divide by a negative

There is one very important difference between equations and inequalities.

Common Mistake

Dividing or multiplying by a negative

If you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.

For example, if you get $- 3 x < 12$ , dividing by -3 gives $x > - 4$ , not $x < - 4$ .

Example

Solving when the coefficient is negative

Solve $5 > 17 - 3 x$ .

Subtract 17 from both sides.
$- 12 > - 3 x$
Divide both sides by -3, remembering to reverse the inequality sign.
$4 < x$
Rewrite it in the more familiar order.
$x > 4$

Common Mistake

Forgetting to reverse the sign

The most common error is going from $- 12 > - 3 x$ to $4 > x$ . Because you divided by -3, the sign must reverse.

Double inequalities

A double inequality puts the variable between two bounds, such as $−2<x≤5-2<x\le 5$ .

This means both conditions must be true:

$x$ is greater than -2
$x$ is less than or equal to 5

Example

Drawing a double inequality

Show $−4<n≤3-4<n\le 3$ on a number line.

Mark -4 with an open circle because $n$ is greater than -4, not equal to -4.
Mark 3 with a filled circle because $n$ can equal 3.
Draw a thick line between -4 and 3.

Example

Listing integer values from a double inequality

$y$ is an integer and $2≤y<72\le y<7$ . List the possible values of $y$ .

The inequality starts at 2, and 2 is included.
The inequality ends before 7, so 7 is not included.
List the integers from 2 to 6.
The possible values are $y = 2, 3, 4, 5, 6$ .

Solving double inequalities

To solve a double inequality, do the same operation to all three parts: the left side, the middle, and the right side.

Example

Solving a double inequality for integers

$y$ is an integer and $3≤2y−1<113\le 2y-1<11$ . List all possible values of $y$ .

Add 1 to all three parts.
$4≤2y<124\le 2y<12$
Divide all three parts by 2.
$2≤y<62\le y<6$
Since $y$ is an integer, list the whole numbers from 2 up to, but not including, 6.
The possible values are $y = 2, 3, 4, 5$ .

Tip

Work on all three parts

For a double inequality, avoid only changing the middle expression. If you add 1 to the middle, you must add 1 to the left and the right as well.

Inequalities with brackets

Sometimes you must expand brackets before solving. Expanding means multiplying the term outside the bracket by every term inside the bracket.

Example

Solving an inequality with brackets

Solve $2 (3 n - 4) > 10$ .

Expand the bracket.
$6 n - 8 > 10$
Add 8 to both sides.
$6 n > 18$
Divide both sides by 6.
$n > 3$

Example

Solving when the variable is on both sides

Solve $5x−6≤2x+95x-6\le 2x+9$ .

Subtract $2 x$ from both sides.
$3x−6≤93x-6\le 9$
Add 6 to both sides.
$3x≤153x\le 15$
Divide both sides by 3.
$x≤5x\le 5$

Exam technique

In the exam

Decide whether each endpoint is included: open circle for $<$ or $>$ , filled circle for $≤\le$ or $≥\ge$ .
When solving, show each inverse operation clearly so the examiner can award method marks.
If you multiply or divide by a negative number, reverse the inequality sign immediately.

Self review

Check yourself

Can you explain the difference between $x < 3$ and $x≤3x\le 3$ on a number line?
If $n$ is an integer and $−2<n≤4-2<n\le 4$ , which values of $n$ are possible?
When solving $10 - 2 x < 4$ , why does the inequality sign change?

Inequalities

What you'll learn

Inequalities are like equations with a range of answers

Showing inequalities on a number line

Listing integer values

Solving one-step and two-step inequalities

The sign changes when you divide by a negative

Double inequalities

Solving double inequalities

Inequalities with brackets

Recap questions

You've reached the end

Inequalities

What you'll learn

Inequalities are like equations with a range of answers

Showing inequalities on a number line

Listing integer values

Solving one-step and two-step inequalities

The sign changes when you divide by a negative

Double inequalities

Solving double inequalities

Inequalities with brackets

Recap questions

You've reached the end

Inequalities

What you'll learn

Inequalities are like equations with a range of answers

Showing inequalities on a number line

Listing integer values

Solving one-step and two-step inequalities

The sign changes when you divide by a negative

Double inequalities

Solving double inequalities

Inequalities with brackets

Recap questions

You've reached the end

Inequalities Revision Guide

Inequalities

What you'll learn

Inequalities are like equations with a range of answers

Showing inequalities on a number line

Listing integer values

Solving one-step and two-step inequalities

The sign changes when you divide by a negative

Double inequalities

Solving double inequalities

Inequalities with brackets

Recap questions

You've reached the end

Inequalities Revision Guide