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], we'll break it down into simple steps:

1. Isolate the Absolute Value Expression:

Start by isolating the absolute value by subtracting 6 from both sides:

[tex]\[
|x - 4| + 6 = 17 \\
|x - 4| = 17 - 6 \\
|x - 4| = 11
\][/tex]

2. Create Two Equations to Remove the Absolute Value:

The expression [tex]\( |x - 4| = 11 \)[/tex] implies two possible equations:

- Case 1: [tex]\( x - 4 = 11 \)[/tex]
- Case 2: [tex]\( x - 4 = -11 \)[/tex]

3. Solve Each Equation:

Case 1:

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

Add 4 to both sides:

[tex]\[
x = 11 + 4 \\
x = 15
\][/tex]

Case 2:

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

Add 4 to both sides:

[tex]\[
x = -11 + 4 \\
x = -7
\][/tex]

4. Combine the Solutions:

The solutions to the equation are [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].

Thus, the correct choice is:

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

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