92. Which number line below shows the solution set for [tex]x-3 \leq y \leq 2x+5[/tex] when [tex]y=1[/tex]?

A. (Number line A)
B. (Number line B)
C. (Number line C)
D. (Number line D)



Answer :

Let's analyze the inequality given for [tex]\( x \)[/tex]:

[tex]\[ x - 3 \leq y \leq 2x + 5 \][/tex]

Given [tex]\( y = 1 \)[/tex], we substitute [tex]\( y \)[/tex] in the inequalities:

1. For the lower bound:
[tex]\[ x - 3 \leq 1 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ x \leq 1 + 3 \][/tex]
[tex]\[ x \leq 4 \][/tex]

2. For the upper bound:
[tex]\[ 1 \leq 2x + 5 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ 1 - 5 \leq 2x \][/tex]
[tex]\[ -4 \leq 2x \][/tex]
[tex]\[ -2 \leq x \][/tex]

So, combining both inequalities, we have:

[tex]\[ -2 \leq x \leq 4 \][/tex]

This means the solution set for [tex]\( x \)[/tex] is the interval [tex]\([-2, 4]\)[/tex].

Therefore, the correct number line should indicate that [tex]\( x \)[/tex] can take any value from [tex]\(-2\)[/tex] to [tex]\(4\)[/tex], inclusive. The correct number line would show a shaded region between [tex]\(-2\)[/tex] and [tex]\(4\)[/tex] inclusive with filled circles at [tex]\(-2\)[/tex] and [tex]\(4\)[/tex].

Make sure to match this interval with the provided options (E), (F), (G), or (H) to find the correct representation.