Let's solve the given compound inequality step-by-step:
[tex]\[
-1 \leq \frac{2x + 3}{3} < 4
\][/tex]
We will break this compound inequality into two separate inequalities and solve each one individually.
First Inequality:
[tex]\[
-1 \leq \frac{2x + 3}{3}
\][/tex]
1. Multiply both sides by 3 to clear the fraction:
[tex]\[
3(-1) \leq 2x + 3 \Rightarrow -3 \leq 2x + 3
\][/tex]
2. Subtract 3 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-3 - 3 \leq 2x \Rightarrow -6 \leq 2x
\][/tex]
3. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-6}{2} \leq x \Rightarrow -3 \leq x
\][/tex]
So the solution to the first inequality is:
[tex]\[
-3 \leq x
\][/tex]
Second Inequality:
[tex]\[
\frac{2x + 3}{3} < 4
\][/tex]
1. Multiply both sides by 3 to clear the fraction:
[tex]\[
3 \left( \frac{2x + 3}{3} \right) < 3 \times 4 \Rightarrow 2x + 3 < 12
\][/tex]
2. Subtract 3 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x + 3 - 3 < 12 - 3 \Rightarrow 2x < 9
\][/tex]
3. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{2x}{2} < \frac{9}{2} \Rightarrow x < 4.5
\][/tex]
So the solution to the second inequality is:
[tex]\[
x < 4.5
\][/tex]
Now, combining the solutions from both inequalities, we get:
[tex]\[
-3 \leq x < 4.5
\][/tex]
In interval notation, we write this solution as:
[tex]\[
[-3, \frac{9}{2})
\][/tex]
This interval notation matches the selection from option d.
So, the solution set in interval notation is:
[tex]\[
\boxed{d. [-3, \frac{9}{2})}
\][/tex]
To graph the solution set, you can draw a number line and shade the region from [tex]\(-3\)[/tex] (inclusive, indicated by a closed dot or bracket) to [tex]\(4.5\)[/tex] (exclusive, indicated by an open dot or parenthesis).
