Solve Simple and Compound Linear Inequalities

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: $-\frac{5}{6} x \leq \frac{3}{4}+\frac{8}{3} x$.