To solve the inequality [tex]\( |4x + 4| - 13 \leq -5 \)[/tex], let's follow a step-by-step method:
1. Start with the inequality:
[tex]\[
|4x + 4| - 13 \leq -5
\][/tex]
2. Isolate the absolute value expression:
[tex]\[
|4x + 4| - 13 \leq -5
\][/tex]
Add 13 to both sides:
[tex]\[
|4x + 4| \leq -5 + 13
\][/tex]
Simplify the right-hand side:
[tex]\[
|4x + 4| \leq 8
\][/tex]
3. Understand the absolute value inequality:
The inequality [tex]\( |A| \leq B \)[/tex] means that [tex]\( -B \leq A \leq B \)[/tex].
4. Apply this property to our inequality:
[tex]\[
-8 \leq 4x + 4 \leq 8
\][/tex]
5. Solve the compound inequality:
Split the inequality into two parts and solve each part:
[tex]\[
-8 \leq 4x + 4 \quad \text{and} \quad 4x + 4 \leq 8
\][/tex]
6. Solve the left-hand inequality:
[tex]\[
-8 \leq 4x + 4
\][/tex]
Subtract 4 from both sides:
[tex]\[
-12 \leq 4x
\][/tex]
Divide by 4:
[tex]\[
-3 \leq x
\][/tex]
7. Solve the right-hand inequality:
[tex]\[
4x + 4 \leq 8
\][/tex]
Subtract 4 from both sides:
[tex]\[
4x \leq 4
\][/tex]
Divide by 4:
[tex]\[
x \leq 1
\][/tex]
8. Combine the solutions:
The combined solution is:
[tex]\[
-3 \leq x \leq 1
\][/tex]
9. Graph the solution:
On a number line, mark the interval [tex]\([-3, 1]\)[/tex]:
- Place a filled circle at [tex]\( x = -3 \)[/tex] to indicate that [tex]\( x = -3 \)[/tex] is included in the solution.
- Place a filled circle at [tex]\( x = 1 \)[/tex] to indicate that [tex]\( x = 1 \)[/tex] is included in the solution.
- Shade the region between [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex].
The solution to the inequality [tex]\( |4x + 4| - 13 \leq -5 \)[/tex] is [tex]\( -3 \leq x \leq 1 \)[/tex].
