To solve the inequality [tex]\(|3x + 7| \leq 4\)[/tex], we need to understand the properties of absolute values. Specifically, the inequality [tex]\(|A| \leq B\)[/tex] is equivalent to [tex]\(-B \leq A \leq B\)[/tex] for any real numbers [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
In this case, [tex]\(A = 3x + 7\)[/tex] and [tex]\(B = 4\)[/tex]. So, we can write the inequality as:
[tex]\[
-4 \leq 3x + 7 \leq 4
\][/tex]
Next, we solve this compound inequality.
1. First Part: [tex]\(-4 \leq 3x + 7\)[/tex]
Subtract 7 from both sides:
[tex]\[
-4 - 7 \leq 3x
\][/tex]
[tex]\[
-11 \leq 3x
\][/tex]
Divide both sides by 3:
[tex]\[
-\frac{11}{3} \leq x
\][/tex]
2. Second Part: [tex]\(3x + 7 \leq 4\)[/tex]
Subtract 7 from both sides:
[tex]\[
3x \leq 4 - 7
\][/tex]
[tex]\[
3x \leq -3
\][/tex]
Divide both sides by 3:
[tex]\[
x \leq -1
\][/tex]
Combining these two results, we get the solution to the inequality:
[tex]\[
-\frac{11}{3} \leq x \leq -1
\][/tex]
So, the solution set for the inequality [tex]\(|3x + 7| \leq 4\)[/tex] is:
[tex]\[
-\frac{11}{3} \leq x \leq -1
\][/tex]
