To solve the compound inequality [tex]\(-5 < 4x + 3 \leq 7\)[/tex], we will break it down into two parts and solve each part separately, then combine the results.
### Step 1: Solve [tex]\(-5 < 4x + 3\)[/tex]
1. Start with the inequality: [tex]\(-5 < 4x + 3\)[/tex]
2. Subtract 3 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
-5 - 3 < 4x
\][/tex]
3. Simplify the left side:
[tex]\[
-8 < 4x
\][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
-8 / 4 < x \Rightarrow -2 < x
\][/tex]
### Step 2: Solve [tex]\(4x + 3 \leq 7\)[/tex]
1. Start with the inequality: [tex]\(4x + 3 \leq 7\)[/tex]
2. Subtract 3 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
4x + 3 - 3 \leq 7 - 3
\][/tex]
3. Simplify the right side:
[tex]\[
4x \leq 4
\][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
4 / 4 \leq x \Rightarrow x \leq 1
\][/tex]
### Step 3: Combine the results
Putting the results from the two parts together:
[tex]\[
-2 < x \quad \text{and} \quad x \leq 1
\][/tex]
Therefore, the solution to the inequality [tex]\(-5 < 4x + 3 \leq 7\)[/tex] is:
[tex]\[
-2 < x \leq 1
\][/tex]
Among the given options, the one that correctly represents this solution is:
[tex]\[
\text{C. } x > -2 \text{ and } x \leq 1
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
