Question

solve the inequality -2 < 3m - 5 ≤ 7 and graph the solution set.
1 ≤ m ≤ 4
○ number line with closed circles at 1 and 4
1 < m ≤ 4
○ number line with open circle at 1 and closed circle at 4
m < 1 or m ≥ 4
● number line with open circle at 1 and closed circle at 4, arrows pointing left from 1 and right from 4
m ≤ 1 or m > 4
○ number line with closed circle at 1 and open circle at 4, arrows pointing left from 1 and right from 4

Explanation:

Step1: Add 5 to all parts

To solve the compound inequality \(-2 < 3m - 5 \leq 7\), we first add 5 to each part of the inequality. This is done to isolate the term with \(m\).

$$ -2 + 5 < 3m - 5 + 5 \leq 7 + 5 $$

Simplifying each part:

$$ 3 < 3m \leq 12 $$

Step2: Divide all parts by 3

Next, we divide each part of the inequality by 3 to solve for \(m\).

$$ \frac{3}{3} < \frac{3m}{3} \leq \frac{12}{3} $$

Simplifying each part gives:

$$ 1 < m \leq 4 $$

For the graph, an open circle at \(m = 1\) (since \(m > 1\), not including 1) and a closed circle at \(m = 4\) (since \(m \leq 4\), including 4), with a line connecting them between 1 and 4.

Answer:

\(1 < m \leq 4\) (and the corresponding graph with an open circle at 1, closed circle at 4, and line between them)