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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]
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]
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