1Cademy - Example: Solving and Graphing One-Step Linear Inequalities

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Linear Inequality

Example

Example: Solving and Graphing One-Step Linear Inequalities

To solve and graph one-step linear inequalities, apply the properties of inequality to isolate the variable, then represent the solution set visually and algebraically.

For example, consider the following three inequalities:

Solve $x - \frac{3}{8} \leq \frac{3}{4}$ : Add $\frac{3}{8}$ to both sides (Addition Property of Inequality). This simplifies to $x \leq \frac{9}{8}$ . On a number line, this is graphed with a right bracket at $\frac{9}{8}$ and shaded to the left. In interval notation, the solution is $(-\infty, \frac{9}{8}]$ .
Solve $9y < 54$ : Divide both sides by $9$ (Division Property of Inequality). Because $9$ is positive, the inequality stays the same, yielding $y < 6$ . This is graphed with a right parenthesis at $6$ and shaded to the left. The solution in interval notation is $(-\infty, 6)$ .
Solve $-15 < \frac{3}{5}z$ : Multiply both sides by $\frac{5}{3}$ (Multiplication Property of Inequality). Since $\frac{5}{3}$ is positive, the inequality sign remains the same, resulting in $-25 < z$ . To rewrite it with the variable on the left, reverse the entire statement: $z > -25$ (Equivalence of Reversed Inequalities). This is graphed with a left parenthesis at $-25$ and shaded to the right. The solution in interval notation is $(-25, \infty)$ .

Updated 2026-04-22

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OpenStax Intermediate Algebra

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