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.
