High School

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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], let's follow a step-by-step approach:

1. Isolate the absolute value expression:

First, we need to isolate the absolute value term on one side of the equation:
[tex]\[
|x - 4| + 6 = 17
\][/tex]
Subtract 6 from both sides:
[tex]\[
|x - 4| = 11
\][/tex]

2. Solve the absolute value equation:

Recall that the absolute value equation [tex]\( |A| = B \)[/tex] implies two possible equations:
[tex]\[
A = B \quad \text{or} \quad A = -B
\][/tex]
For our equation [tex]\( |x - 4| = 11 \)[/tex], this means:
[tex]\[
x - 4 = 11 \quad \text{or} \quad x - 4 = -11
\][/tex]

3. Solve each case separately:

- For [tex]\( x - 4 = 11 \)[/tex]:
[tex]\[
x = 11 + 4
\][/tex]
[tex]\[
x = 15
\][/tex]

- For [tex]\( x - 4 = -11 \)[/tex]:
[tex]\[
x = -11 + 4
\][/tex]
[tex]\[
x = -7
\][/tex]

4. List the solutions:

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

Comparing these solutions to the provided answer choices, we see:

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

Thus, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]

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