1Cademy - Example: Solving and Graphing $$3(2x+5) \leq 18$$ and $$2(x-7)

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Procedure for Solving Compound Inequalities with "and"

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Example: Solving and Graphing $3(2x+5) \leq 18$ and $2(x-7) < -6$

To solve the compound inequality $3(2x+5) \leq 18$ and $2(x-7) < -6$ , start by simplifying both sides of each inequality. For the first inequality, expanding the left side gives $6x + 15 \leq 18$ . Subtracting $15$ yields $6x \leq 3$ , which simplifies to $x \leq \frac{1}{2}$ . For the second inequality, expanding gives $2x - 14 < -6$ . Adding $14$ results in $2x < 8$ , which simplifies to $x < 4$ . Graphing each solution shows that the values satisfying both conditions must be in the overlap. The graph of $x \leq \frac{1}{2}$ is shaded to the left with a right bracket at $\frac{1}{2}$ , and $x < 4$ is shaded to the left with a right parenthesis at $4$ . The intersection of these two graphs is simply the region where $x \leq \frac{1}{2}$ , because any number less than or equal to $\frac{1}{2}$ is also automatically less than $4$ . In interval notation, this final combined solution is written as $(-\infty, \frac{1}{2}]$ .

Updated 2026-06-18

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OpenStax Intermediate Algebra

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