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

Answer :

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

1. Isolate the Absolute Value:
Start by subtracting 6 from both sides to isolate the absolute value:

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

Simplifying gives:

[tex]\[
|x - 4| = 11
\][/tex]

2. Solve the Absolute Value Equation:
The equation [tex]\( |x - 4| = 11 \)[/tex] means that the expression inside the absolute value can either be 11 or -11. So we have two cases to consider:

Case 1: [tex]\( x - 4 = 11 \)[/tex]

- Add 4 to both sides:

[tex]\[
x = 11 + 4
\][/tex]

- So, [tex]\( x = 15 \)[/tex].

Case 2: [tex]\( x - 4 = -11 \)[/tex]

- Add 4 to both sides:

[tex]\[
x = -11 + 4
\][/tex]

- So, [tex]\( x = -7 \)[/tex].

3. Solution:
The solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].

Thus, the correct answer is option C: [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].

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