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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 :

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

1. Isolate the absolute value expression:

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

Subtract 6 from both sides to isolate the absolute value term:

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

2. Consider the definition of absolute value:

The absolute value [tex]\( |x - 4| = 11 \)[/tex] means that [tex]\( x - 4 \)[/tex] can be either 11 or -11. Therefore, we have two cases to consider:

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

3. Solve each case separately:

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

Add 4 to both sides:

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

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

Add 4 to both sides:

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

4. Combine the solutions:

So, the solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are:

[tex]\[
x = 15 \quad \text{and} \quad x = -7
\][/tex]

Checking the given options, we can see that the correct answer is:

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

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