Question

Solve the inequality and graph the solution. $$ 3 x+8>17 \text { or } 2 x+5 \leq 7 $$

Short answer

The solution for the given inequalities are \( x \leq 1 \) or \( x > 3 \). This includes all numbers less than or equal to 1 or greater than 3.

Step-by-step solution

Solve the first inequality

Subtract 8 from both sides of the inequality \( 3x+8>17 \). This gives \( 3x > 17 - 8 = 9 \). Then, divide each side of the inequality by 3, obtaining \( x>3 \).

Solve the second inequality

Subtract 5 from both sides of the inequality \( 2x+5\leq7 \) to get \( 2x \leq 7 - 5 = 2 \). Then, divide each side of the inequality by 2 to get \( x \leq 1 \).

Graph the solutions

Our solutions for the inequalities are \( x>3 \) and \( x \leq 1 \). We graph them separately. Number 3 will be represented with an open dot on the number line and everything greater than 3 in the positive direction will be shaded. The number 1 will be represented with a filled dot on the number line and everything less than or equal to 1 in the negative direction will be shaded.

Interpret the final graph

Because the statement was 'or', the solution is any \( x \) that satisfies either inequality. From the graph, we can see that this includes all \( x \) less than or equal to 1 and all \( x \) greater than 3.