College

Solve for [tex]\( x \)[/tex].

[tex]\[ |x-9| \leq 7 \][/tex]

A. [tex]\( x \leq 2 \)[/tex] or [tex]\( x \geq 16 \)[/tex]

B. [tex]\( -16 \leq x \leq -2 \)[/tex]

C. [tex]\( -7 \leq x \leq 7 \)[/tex]

D. [tex]\( 2 \leq x \leq 16 \)[/tex]

Answer :

To solve the inequality [tex]\( |x - 9| \leq 7 \)[/tex], we need to break it down into two separate inequalities based on the definition of absolute value.

The absolute value inequality [tex]\( |x - 9| \leq 7 \)[/tex] can be rewritten as:
[tex]\[ -7 \leq x - 9 \leq 7 \][/tex]

We will solve for [tex]\( x \)[/tex] by breaking this into two parts:

1. [tex]\( x - 9 \geq -7 \)[/tex]
2. [tex]\( x - 9 \leq 7 \)[/tex]

### Solving the first part:
[tex]\[ x - 9 \geq -7 \][/tex]

To isolate [tex]\( x \)[/tex], add 9 to both sides:
[tex]\[ x - 9 + 9 \geq -7 + 9 \][/tex]
[tex]\[ x \geq 2 \][/tex]

### Solving the second part:
[tex]\[ x - 9 \leq 7 \][/tex]

To isolate [tex]\( x \)[/tex], add 9 to both sides:
[tex]\[ x - 9 + 9 \leq 7 + 9 \][/tex]
[tex]\[ x \leq 16 \][/tex]

### Combining both parts:
From the two inequalities [tex]\( x \geq 2 \)[/tex] and [tex]\( x \leq 16 \)[/tex], we can combine these to get:
[tex]\[ 2 \leq x \leq 16 \][/tex]

This means [tex]\( x \)[/tex] must be between 2 and 16, inclusive.

Therefore, the correct answer is:
[tex]\[ D. \, 2 \leq x \leq 16 \][/tex]