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Solve [tex]|x-4|+6=17[/tex].

A. [tex]x=15[/tex] and [tex]x=-7[/tex]

B. [tex]x=-15[/tex] and [tex]x=-7[/tex]

C. [tex]x=-15[/tex] and [tex]x=7[/tex]

D. [tex]x=15[/tex] and [tex]x=-15[/tex]

Answer :

To solve the equation [tex]\( |x - 4| + 6 = 17 \)[/tex], we need to follow these steps:

1. Isolate the Absolute Value:
[tex]\[
|x - 4| + 6 = 17
\][/tex]
We start by subtracting 6 from both sides of the equation:
[tex]\[
|x - 4| = 17 - 6
\][/tex]
Simplifying the right side, we get:
[tex]\[
|x - 4| = 11
\][/tex]

2. Set Up Two Separate Equations:
The absolute value equation [tex]\( |x - 4| = 11 \)[/tex] means there are two scenarios to consider:
[tex]\[
x - 4 = 11 \quad \text{or} \quad x - 4 = -11
\][/tex]

3. Solve Each Equation:
- For the first equation [tex]\( x - 4 = 11 \)[/tex]:
[tex]\[
x - 4 = 11
\][/tex]
Add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 11 + 4
\][/tex]
Thus:
[tex]\[
x = 15
\][/tex]

- For the second equation [tex]\( x - 4 = -11 \)[/tex]:
[tex]\[
x - 4 = -11
\][/tex]
Add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -11 + 4
\][/tex]
Thus:
[tex]\[
x = -7
\][/tex]

4. Write the Solutions:
The solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are:
[tex]\[
x = 15 \quad \text{and} \quad x = -7
\][/tex]

Therefore, the correct option is:

A. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex]

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