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Inequalities

What you'll learn

  • understand what inequality signs mean
  • list possible whole-number values between two limits
  • draw and read number lines with open and solid circles
  • solve inequalities with letters, brackets, and letters on both sides

1. What an inequality says

Before solving, remember that a letter such as xxx or nnn can stand for a number.

Definition

Inequality, variable and solution

An inequality compares values using signs such as <<<, >>>, ≤\le≤ and ≥\ge≥. A variable is a letter, such as xxx or nnn, that represents a number. A solution is any value that makes the inequality true.

Read the inequality from left to right:

  • x<4x < 4x<4 means xxx is less than 4.
  • x>4x > 4x>4 means xxx is greater than 4.
  • x≤4x \le 4x≤4 means xxx is less than or equal to 4.
  • x≥4x \ge 4x≥4 means xxx is greater than or equal to 4.
Definition

Integers

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

Key Idea

The line means included

The symbols ≤\le≤ and ≥\ge≥ include the end value. The symbols <<< and >>> do not include the end value.

Open circles show excluded endpoints, while solid circles show included endpoints.

Example

Listing possible integer values

A question says: "nnn is an integer and −4<n≤2-4 < n \le 2−4<n≤2." List all possible values of nnn.

The allowed integer values are the whole-number ticks after -4 up to and including 2.

  1. Read the left end, −4<n-4 < n−4<n. This means nnn is bigger than -4, so -4 is not allowed.

  2. Read the right end, n≤2n \le 2n≤2. This means 2 is allowed.

  3. List the integers in order:

    −3, −2, −1, 0, 1, 2-3,\ -2,\ -1,\ 0,\ 1,\ 2−3, −2, −1, 0, 1, 2
Common Mistake

Missing one end

If a question says "integer", do not include decimal values. Also check whether each end value is included or excluded before you write your list.

2. Number lines

A number line shows numbers in order from left to right. Smaller numbers are on the left, larger numbers are on the right.

For inequalities:

  • an open circle means the end value is not included, for <<< or >>>
  • a solid circle means the end value is included, for ≤\le≤ or ≥\ge≥
  • an arrow right means greater than
  • an arrow left means less than
  • a thick line between two points means all values between them
Example

Showing a double inequality

Show −2≤x<3-2 \le x < 3−2≤x<3 on a number line.

The solid circle includes -2, the open circle excludes 3, and the thick line shows all values between them.

  1. At -2, draw a solid circle because xxx can equal -2.

  2. At 3, draw an open circle because xxx is less than 3, not equal to 3.

  3. Join the two circles with a thick line, because every value between them is allowed.

Example

Writing the inequality from a number line

A number line has an open circle at 1 and an arrow pointing left. Write the inequality for xxx.

An open circle at 1 with an arrow to the left represents values less than 1 but not including 1.

  1. The arrow left means values less than 1.

  2. The open circle means 1 is not included.

  3. So the inequality is:

    x<1x < 1x<1

3. Solving inequalities

Solving an inequality is very similar to solving an equation. You undo the operations to get the variable on its own.

Key Idea

Keep it balanced

Whatever you add, subtract, multiply or divide on one side, do the same to the other side. If you only multiply or divide by a positive number, the inequality sign stays the same.

Example

Solving a two-step inequality

Solve 3t+5≤173t + 5 \le 173t+5≤17.

  1. Subtract 5 from both sides:

    3t≤123t \le 123t≤12

  2. Divide both sides by 3:

    t≤4t \le 4t≤4

4. Letters on both sides

Sometimes the variable appears on both sides of the inequality. First, collect the variable parts on one side, then collect the number parts on the other side.

Tip

Make the variable part positive

When possible, subtract the smaller variable part from both sides. This reduces the chance of needing to reverse the sign later.

Example

Solving and showing on a number line

Solve 5x+1≤2x+135x + 1 \le 2x + 135x+1≤2x+13 and show the answer on a number line.

The solution is shown with a solid circle at 4 and an arrow to the left because 4 is included and all smaller values work.

  1. Subtract 2x2x2x from both sides:

    3x+1≤133x + 1 \le 133x+1≤13

  2. Subtract 1 from both sides:

    3x≤123x \le 123x≤12

  3. Divide both sides by 3:

    x≤4x \le 4x≤4

  4. On a number line, draw a solid circle at 4 and an arrow to the left.

5. When the sign reverses

This is the special rule that catches lots of students out.

Common Mistake

Dividing by a negative

If you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign. For example, greater than becomes less than.

Example

Reversing the inequality sign

Solve 5>17−3x5 > 17 - 3x5>17−3x.

After dividing by a negative, the final solution is shown as values greater than 4.

  1. Subtract 17 from both sides:

    −12>−3x-12 > -3x−12>−3x

  2. Divide both sides by -3 and reverse the sign:

    4<x4 < x4<x

  3. Write the variable on the left:

    x>4x > 4x>4
Common Mistake

Flipping too often

The sign does not change when you add or subtract. It only changes when you multiply or divide both sides by a negative number.

6. Brackets and compound inequalities

Expanding brackets first

To expand brackets, multiply everything inside the brackets by the number outside. Then solve as normal.

Example

Solving an inequality with brackets

Solve 3(n+2)<213(n + 2) < 213(n+2)<21.

  1. Expand the brackets:

    3n+6<213n + 6 < 213n+6<21

  2. Subtract 6 from both sides:

    3n<153n < 153n<15

  3. Divide both sides by 3:

    n<5n < 5n<5

Compound inequalities

Definition

Compound inequalities

A compound inequality has two inequality signs. It tells you the variable must fit between two end values, like −2<x≤5-2 < x \le 5−2<x≤5.

Do the same operation to all three parts.

Example

Solving a compound inequality with integers

Let yyy be an integer and 3≤2y−1<113 \le 2y - 1 < 113≤2y−1<11. List the possible values of yyy.

The solved compound inequality includes 2, excludes 6, and the integer values lie at the whole-number ticks between them.

  1. Add 1 to all three parts:

    4≤2y<124 \le 2y < 124≤2y<12

  2. Divide all three parts by 2:

    2≤y<62 \le y < 62≤y<6

  3. Since yyy is an integer, list the whole numbers allowed:

    2, 3, 4, 52,\ 3,\ 4,\ 52, 3, 4, 5

  4. On a number line, this would be a solid circle at 2, an open circle at 6, and a thick line between them.

Exam technique

In the exam

  1. Check whether the question wants a solved inequality, a number line, or a list of integer values.

  2. For number lines, decide open circle or solid circle before drawing the arrow or line segment.

  3. If you divide or multiply by a negative number, reverse the inequality sign.

Self review

Check yourself

  • Can you explain the difference between x<4x < 4x<4 and x≤4x \le 4x≤4 on a number line?

  • If nnn is an integer and −2<n≤3-2 < n \le 3−2<n≤3, which values are possible?

  • In 7>13−2x7 > 13 - 2x7>13−2x, where does the inequality sign change?

Recap questions

1 of 5

Let nnn be an integer. Which list gives all possible values of nnn if −3<n≤2-3 < n \le 2−3<n≤2?

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Inequalities Revision Guide

  1. GCSE
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