Question

Solve the absolute value inequality.
StartAbsoluteValue x plus 6 EndAbsoluteValue less than or equals 9
Question content area bottom
Part 1
Rewrite the inequality without absolute value bars. Select the correct choice below and fill in the answer​ box(es) to complete your choice.
​(Simplify your​ answers.)
A.
xplus6less than or equals
  
enter your response here or xplus6greater than or equals
  
enter your response here
B.
  
enter your response hereless than or equalsxplus6less than or equals
  
enter your response here
C.
xplus6less than or equals
  
enter your response here

Answer 1

To solve the absolute value inequality \( |x + 6| \leq 9 \), we need to rewrite it without the absolute value bars. The absolute value inequality \( |A| \leq B \) translates to:

\[
-B \leq A \leq B
\]

in this case, where \( A = x + 6 \) and \( B = 9 \).

So, we can rewrite the inequality as:

\[
-9 \leq x + 6 \leq 9
\]

### Now we solve this compound inequality:

1. **For the left side**:

\[
-9 \leq x + 6
\]
Subtract 6 from both sides:
\[
-9 - 6 \leq x
\]
\[
-15 \leq x
\]

2. **For the right side**:

\[
x + 6 \leq 9
\]
Subtract 6 from both sides:
\[
x \leq 9 - 6
\]
\[
x \leq 3
\]

### Combining both parts, we have:

\[
-15 \leq x \leq 3
\]

Thus, we can write the final solution as:

\[
-15 \leq x \leq 3
\]

### Choosing the correct option:

Based on the options provided:

**Part 1** should be option B:

\[
-15 \leq x \leq 3
\]

### So your answers to fill in the options are:

Option B would be filled as:

- Enter the response for the left side: **-15**
- Enter the response for the right side: **3**