Question

solve the inequality $-2 < 3m - 5 \leq 7$ and graph the solution set.
$1 \leq m \leq 4$
$\circ$ \

$$\begin{tikzpicture}>=stealth,scale=0.6 \\draw-> (-10,0) -- (10,0); \\foreach \\x in {-10,-9,...,10} \\draw (\\x,0.2) -- (\\x,-0.2) nodebelow {$\\x$}; \\fillblack (1,0) circle (2pt); \\fillblack (4,0) circle (2pt); \\drawthick (1,0) -- (4,0); \\end{tikzpicture}$$

$1 < m \leq 4$
$\circ$ \

$$\begin{tikzpicture}>=stealth,scale=0.6 \\draw-> (-10,0) -- (10,0); \\foreach \\x in {-10,-9,...,10} \\draw (\\x,0.2) -- (\\x,-0.2) nodebelow {$\\x$}; \\drawthick, fill=white (1,0) circle (2pt); \\fillblack (4,0) circle (2pt); \\drawthick (1,0) -- (4,0); \\end{tikzpicture}$$

$m < 1$ or $m \geq 4$
$\circ$ \

$$\begin{tikzpicture}>=stealth,scale=0.6 \\draw-> (-10,0) -- (10,0); \\foreach \\x in {-10,-9,...,10} \\draw (\\x,0.2) -- (\\x,-0.2) nodebelow {$\\x$}; \\drawthick, fill=white (1,0) circle (2pt); \\fillblack (4,0) circle (2pt); \\drawthick (-10,0) -- (1,0); \\drawthick (4,0) -- (10,0); \\end{tikzpicture}$$

$m \leq 1$ or $m > 4$
$\circ$ \

$$\begin{tikzpicture}>=stealth,scale=0.6 \\draw-> (-10,0) -- (10,0); \\foreach \\x in {-10,-9,...,10} \\draw (\\x,0.2) -- (\\x,-0.2) nodebelow {$\\x$}; \\fillblack (1,0) circle (2pt); \\drawthick, fill=white (4,0) circle (2pt); \\drawthick (-10,0) -- (1,0); \\drawthick (4,0) -- (10,0); \\end{tikzpicture}$$

Explanation:

Step1: Solve the inequality

We start with the compound inequality \(-2 < 3m - 5 \leq 7\). First, we add 5 to all parts of the inequality to isolate the term with \(m\).
Adding 5 to each part: \(-2 + 5 < 3m - 5 + 5 \leq 7 + 5\)
Simplifying each part: \(3 < 3m \leq 12\)
Then, we divide all parts by 3 to solve for \(m\): \(\frac{3}{3} < \frac{3m}{3} \leq \frac{12}{3}\)
Simplifying gives: \(1 < m \leq 4\)

Step2: Analyze the graph

For the inequality \(1 < m \leq 4\), the left - hand side is a strict inequality (\(<\)), so we use an open circle at \(m = 1\) to indicate that 1 is not included in the solution set. The right - hand side is a non - strict inequality (\(\leq\)), so we use a closed circle at \(m = 4\) to indicate that 4 is included in the solution set. Then we draw a line segment between the open circle at 1 and the closed circle at 4 to represent all the values of \(m\) that satisfy the inequality.

Answer:

The solution to the inequality \(-2 < 3m - 5 \leq 7\) is \(1 < m \leq 4\), and the corresponding graph is the one with an open circle at 1 and a closed circle at 4, with a line connecting them (the second graph option). So the answer is the option with \(1 < m \leq 4\) and the graph with an open circle at 1 and a closed circle at 4.