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

Answer :

Sure! Let's solve the equation step-by-step:

We have the equation:
[tex]\[ |x - 4| + 6 = 17 \][/tex]

First, we need to isolate the absolute value:
[tex]\[ |x - 4| = 17 - 6 \][/tex]
[tex]\[ |x - 4| = 11 \][/tex]

Next, we need to solve the absolute value equation. Recall that if [tex]\(|A| = B\)[/tex], then [tex]\(A = B\)[/tex] or [tex]\(A = -B\)[/tex]. Therefore:
[tex]\[ x - 4 = 11 \][/tex]
or
[tex]\[ x - 4 = -11 \][/tex]

Now solve these two simple equations:
1. For [tex]\(x - 4 = 11\)[/tex]:
[tex]\[ x - 4 = 11 \][/tex]
[tex]\[ x = 11 + 4 \][/tex]
[tex]\[ x = 15 \][/tex]

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

So, the solutions are [tex]\(x = 15\)[/tex] and [tex]\(x = -7\)[/tex].

Looking at the given choices:

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

The correct answer is:
[tex]\[ \boxed{D. \, x = 15 \text{ and } x = -7} \][/tex]

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