Question

solve the compound inequality. give the solution set in both interval and graph forms
$x - 2 \leq 2$ and $x + 1 \geq 3$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the solution set is \\(\square\\)
\\(\quad\\) (type your answer in interval notation.)
\\(\bigcirc\\) b. the solution set is \\(\varnothing\\)

choose the correct graph form below.
\\(\bigcirc\\) a. \\(\longleftarrow\hspace{-0.8em}\stackrel{\
ightharpoonup}{\hspace{1.5em}}\

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c}0&1&\\boldsymbol{\\mathbf{2}}&3&4&5&6&7&8&9&10\\end{array}$$

\hspace{-0.8em}\stackrel{\leftharpoonup}{\hspace{1.5em}}\longrightarrow\\) \\(\bigcirc\\) b. \\(\longleftarrow\hspace{-0.8em}\stackrel{\
ightharpoonup}{\hspace{1.5em}}\

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c}0&1&\\boldsymbol{\\mathbf{2}}&3&4&5&\\boldsymbol{\\mathbf{6}}&7&8&9&10\\end{array}$$

\hspace{-0.8em}\stackrel{\leftharpoonup}{\hspace{1.5em}}\longrightarrow\\)
\\(\bigcirc\\) c. \\(\longleftarrow\hspace{-0.8em}\stackrel{\
ightharpoonup}{\hspace{1.5em}}\

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c}0&1&\\boldsymbol{\\mathbf{2}}&3&\\boldsymbol{\\mathbf{4}}&5&6&7&8&9&10\\end{array}$$

\hspace{-0.8em}\stackrel{\leftharpoonup}{\hspace{1.5em}}\longrightarrow\\) \\(\bigcirc\\) d. \\(\longleftarrow\hspace{-0.8em}\stackrel{\
ightharpoonup}{\hspace{1.5em}}\

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c}0&1&\\boldsymbol{\\mathbf{2}}&3&\\boldsymbol{\\mathbf{4}}&5&6&7&8&9&10\\end{array}$$

\hspace{-0.8em}\stackrel{\leftharpoonup}{\hspace{1.5em}}\longrightarrow\\)
\\(\bigcirc\\) e. \\(\longleftarrow\hspace{-0.8em}\stackrel{\
ightharpoonup}{\hspace{1.5em}}\

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c}0&1&2&3&4&5&6&7&8&9&10\\end{array}$$

\hspace{-0.8em}\stackrel{\leftharpoonup}{\hspace{1.5em}}\longrightarrow\\) \\(\bigcirc\\) f. \\(\longleftarrow\hspace{-0.8em}\stackrel{\
ightharpoonup}{\hspace{1.5em}}\

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c}0&1&2&3&4&5&6&7&8&9&10\\end{array}$$

\hspace{-0.8em}\stackrel{\leftharpoonup}{\hspace{1.5em}}\longrightarrow\\)

Explanation:

Step1: Solve \( x - 2 \leq 2 \)

Add 2 to both sides of the inequality:
\( x - 2 + 2 \leq 2 + 2 \)
\( x \leq 4 \)

Step2: Solve \( x + 1 \geq 3 \)

Subtract 1 from both sides of the inequality:
\( x + 1 - 1 \geq 3 - 1 \)
\( x \geq 2 \)

Step3: Find the intersection

The solution to the compound inequality (since it is "and") is the overlap of \( x \leq 4 \) and \( x \geq 2 \), which is \( 2 \leq x \leq 4 \). In interval notation, this is \([2, 4]\).

For the graph, we look for a segment from 2 to 4 with closed circles (since the inequality includes equality), which matches option C.

Answer:

• Interval notation: \(\boldsymbol{[2, 4]}\)
• Graph: Option C (the number line with a closed interval from 2 to 4).

(For the multiple-choice part of the interval: A. The solution set is \(\boldsymbol{[2, 4]}\); for the graph: C.)