# Inequalities: Greater than or less than

## In a nutshell

Inequalities show when one expression is greater than or less than another expression. Inequalities with more complex expressions can be solved in a similar way to equations. The answers can be represented algebraically or on a number line.

## Symbols

Inequalities involve using symbols to describe the relationship between two expressions.

**SYMBOL** | **DESCRIPTION** |

$\lt$ | less than |

$\le$ | less than or equal to |

$\gt$ | greater than |

$\ge$ | greater than or equal to |

**Note:*** The arrow points to the smaller number.*

## Number lines

$x \le 3$ means that $x$ is less than or equal to $3$. It can be represented by the number line

use $\circ$ for $\gt or \lt$

use $\bullet$ for $\ge or \le$

## Solving inequalities

You can use knowledge of rearranging equations to solve the inequalities.

### Basic Inequalities

Rearrange like equations to solve.

##### Example 1

$\begin {aligned}3x + 10 &\le 22 \\3x &\le 12 \\ \end {aligned}\\ \qquad \underline{x \le 4}$

### Inequalities with negatives

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

##### Example 2

$\begin {aligned}-2x &\lt 10 \\ & \qquad \div -2\\ \end {aligned}\\ \quad \underline {x \gt 5}$

### Inequalities in two parts

Like solving an equation, do the same to each of the 3 parts of the inequality.

##### Example 3

$\begin {aligned}21 &\lt 4x-3 &\lt 41 \\&&& +3 \\24 &\lt \quad 4x &\lt 44 \\&&& \div 4 \\\end {aligned}\\ \quad \underline{6 \lt x \lt 11}$

### Complex Inequalities in two parts

Split the inequality up into two separate questions, solve each separately then recombine the answers.

##### Example 4

$\begin {aligned}&&3x-19 &\lt 5x-3 &\lt 4x+2 \\3x-19 &\lt 5x-3 &&& 5x-3 &\lt 4x+2\\-8 &\lt x &&& x &\lt 5\\\end {aligned}$

$\underline {-8 \lt x \lt 5}$