Linear Inequalities (Single & Double Inequalities)

What is a Linear Inequality?

A linear inequality is like a linear equation, but instead of using $=$,  it uses inequality symbols like:

• $\gt$ (greater than)
• $\lt$ (less than)
• $\ge$ (greater than or equal to)
• $\le$ (less than or equal to)

 

Examples of Linear Inequalities:

• $2x + 3 \gt 7$
• $5x - 4 \le 16$
• $-3x + 2 \ge 5$

The solution to an inequality is a range of values that make the inequality true.

 

Key Rules for Solving Linear Inequalities

1. Treat the inequality like an equation. Perform the same operations (add, subtract, multiply, divide) on both sides.
2. Flip the inequality sign if you multiply or divide by a negative number. Example: $$-2x < 6 \quad \text{Divide both sides by } -2 \\ x > -3$$
   
   
3. For double inequalities, solve the inequality in sections. Example: $$3 < 2x - 1 \leq 7$$

 

Graphing Solutions

When graphing inequalities on a number line:

• Use an open circle for $\lt$ or $\gt$ (not including the value)

• Use a closed circle for $\le$ or $\ge$ (including the value)

• Shade the region showing all possible values of $x$

 

Examples

Single Inequality

Example 1: Solve: $$2x + 5 \leq 11$$

1. Subtract $5$ from both sides: $$2x + 5 - 5 \leq 11 - 5 \\ 2x \leq 6$$
   
   
2. Divide both sides by $2$: $$\frac{2x}{2} \leq \frac{6}{2} \\  x \leq 3$$
   
   

Solution: $x \le 3$

 

 

 

 

 

Double Inequality

Example 2: Solve : $$-2 < 3x + 1 \leq 10$$

1. Subtract $1$ from all parts: $$-2 - 1 < 3x + 1 - 1 \leq 10 - 1 \\  -3 < 3x \leq 9$$
   
   
2. Divide all parts by $3$: $$\frac{-3}{3} < \frac{3x}{3} \leq \frac{9}{3} \\ -1 < x \leq 3$$
   
   

Solution: $-1 \lt x \le 3$