Solve Simple and Compound Linear Inequalities

Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

Example 5

Solving an Inequality Algebraically

Solve the inequality: $13-7 x \geq 10 x-4$ .

Solution

Solving this inequality is similar to solving an equation up until the last step.

$\begin{array}{cl} 13-7 x \geq 10 x-4 & \\ 13-17 x \geq-4 & \text { Move variable terms to one side of the inequality. } \\ -17 x \geq-17 & \text { Isolate the variable term. } \\ x \leq 1 & \text { Dividing both sides by }-17 \text { reverses the inequality. } \end{array}$

The solution set is given by the interval $(-\infty, 1]$ , or all real numbers less than and including $1$ .

Try It #5

Solve the inequality and write the answer using interval notation: $-x+4 < \frac{1}{2} x+1$ .

Example 6

Solving an Inequality with Fractions

Solve the following inequality and write the answer in interval notation: $-\frac{3}{4} x \geq-\frac{5}{8}+\frac{2}{3} x$ .

Solution

We begin solving in the same way we do when solving an equation.

$\begin{aligned} -\frac{3}{4} x & \geq-\frac{5}{8}+\frac{2}{3} x & & \\ -\frac{3}{4} x-\frac{2}{3} x & \geq-\frac{5}{8} & & \text { Put variable terms on one side. } \\ -\frac{9}{12} x-\frac{8}{12} x & \geq-\frac{5}{8} & & \text { Write fractions with common denominator. } \\ -\frac{17}{12} x & \geq-\frac{5}{8} & \\ x & \leq-\frac{5}{8}\left(-\frac{12}{17}\right) & & \text { Multiplying by a negative number reverses the inequality. } \\ x & \leq \frac{15}{34} & & \end{aligned}$

The solution set is the interval $\left(-\infty, \frac{15}{34}\right]$ .

Try It #6

Solve the inequality and write the answer in interval notation: $-\frac{5}{6} x \leq \frac{3}{4}+\frac{8}{3} x$ .