To solve the inequality [tex]\(|3x + 7| \leq 4\)[/tex], we need to split it into two separate inequalities, because the absolute value function splits the number line into two cases:
### Case 1:
[tex]\[ 3x + 7 \leq 4 \][/tex]
To solve this:
1. Subtract 7 from both sides:
[tex]\[ 3x \leq 4 - 7 \][/tex]
[tex]\[ 3x \leq -3 \][/tex]
2. Divide both sides by 3:
[tex]\[ x \leq -1 \][/tex]
### Case 2:
[tex]\[ 3x + 7 \geq -4 \][/tex]
To solve this:
1. Subtract 7 from both sides:
[tex]\[ 3x \geq -4 - 7 \][/tex]
[tex]\[ 3x \geq -11 \][/tex]
2. Divide both sides by 3:
[tex]\[ x \geq -\frac{11}{3} \][/tex]
Now, combining the results of both cases, we get:
[tex]\[ -\frac{11}{3} \leq x \leq -1 \][/tex]
Therefore, the solution to the inequality [tex]\(|3x + 7| \leq 4\)[/tex] is:
[tex]\[ -\frac{11}{3} \leq x \leq -1 \][/tex]
In other words:
[tex]\[ -3.6666666666666665 \leq x \leq -1 \][/tex]
So the final answer is:
[tex]\[ -3.6666666666666665 \leq x \leq -1 \][/tex]
