Solving inequalities
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
$\begin {aligned}21 &\lt 4x3 &\lt 41 \\&&& +3 \\24 &\lt \quad 4x &\lt 44 \\&&& \div 4 \\\end {aligned}\\ \quad \underline{6 \lt x \lt 11}$
Complex Inequalities in two parts
Split the inequality up into 2 separate questions, solve each separately then recombine the answers.
Example 4
$\begin {aligned}&&3x19 &\lt 5x3 &\lt 4x+2 \\3x19 &\lt 5x3 &&& 5x3 &\lt 4x+2\\8 &\lt x &&& x &\lt 5\\\end {aligned}$
$\underline {8 \lt x \lt 5}$