1Cademy - Example: Solving and Graphing $$ 5(3x-1) \leq 10 $$ and $$ 4(x+3)

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Procedure for Solving Compound Inequalities with "and"

Example

Example: Solving and Graphing $5(3x-1) \leq 10$ and $4(x+3) < 8$

To solve the compound inequality $5(3x-1) \leq 10$ and $4(x+3) < 8$ , begin by solving each inequality independently. Distributing the $5$ in the first inequality gives $15x - 5 \leq 10$ , which simplifies to $15x \leq 15$ or $x \leq 1$ . Distributing the $4$ in the second inequality yields $4x + 12 < 8$ , which simplifies to $4x < -4$ or $x < -1$ . Graphing both results on a number line shows that the intersection of $x \leq 1$ and $x < -1$ is the set of all numbers where $x < -1$ . In interval notation, this overlapping solution is written as $(-\infty, -1)$ .