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Answer :
Sure, let's solve the equation [tex]\( |x-4| + 6 = 17 \)[/tex] step-by-step.
1. Isolate the absolute value term:
[tex]\[
|x - 4| + 6 = 17
\][/tex]
Subtract 6 from both sides to isolate the absolute value:
[tex]\[
|x - 4| = 17 - 6
\][/tex]
[tex]\[
|x - 4| = 11
\][/tex]
2. Set up two equations to handle the absolute value:
The absolute value equation [tex]\( |x - 4| = 11 \)[/tex] means that the expression inside the absolute value, [tex]\( x - 4 \)[/tex], can be either [tex]\( 11 \)[/tex] or [tex]\( -11 \)[/tex].
So we have two separate cases:
[tex]\[
x - 4 = 11
\][/tex]
and
[tex]\[
x - 4 = -11
\][/tex]
3. Solve each equation:
- For the first case:
[tex]\[
x - 4 = 11
\][/tex]
Add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 11 + 4
\][/tex]
[tex]\[
x = 15
\][/tex]
- For the second case:
[tex]\[
x - 4 = -11
\][/tex]
Add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -11 + 4
\][/tex]
[tex]\[
x = -7
\][/tex]
4. Write the solution:
The solutions to the equation [tex]\( |x-4| + 6 = 17 \)[/tex] are:
[tex]\[
x = 15 \quad \text{and} \quad x = -7
\][/tex]
So, the correct option from the given choices is C. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
1. Isolate the absolute value term:
[tex]\[
|x - 4| + 6 = 17
\][/tex]
Subtract 6 from both sides to isolate the absolute value:
[tex]\[
|x - 4| = 17 - 6
\][/tex]
[tex]\[
|x - 4| = 11
\][/tex]
2. Set up two equations to handle the absolute value:
The absolute value equation [tex]\( |x - 4| = 11 \)[/tex] means that the expression inside the absolute value, [tex]\( x - 4 \)[/tex], can be either [tex]\( 11 \)[/tex] or [tex]\( -11 \)[/tex].
So we have two separate cases:
[tex]\[
x - 4 = 11
\][/tex]
and
[tex]\[
x - 4 = -11
\][/tex]
3. Solve each equation:
- For the first case:
[tex]\[
x - 4 = 11
\][/tex]
Add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 11 + 4
\][/tex]
[tex]\[
x = 15
\][/tex]
- For the second case:
[tex]\[
x - 4 = -11
\][/tex]
Add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -11 + 4
\][/tex]
[tex]\[
x = -7
\][/tex]
4. Write the solution:
The solutions to the equation [tex]\( |x-4| + 6 = 17 \)[/tex] are:
[tex]\[
x = 15 \quad \text{and} \quad x = -7
\][/tex]
So, the correct option from the given choices is C. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
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