Answer :
To solve the equation [tex]\( |x - 4| + 6 = 17 \)[/tex], let's follow these steps:
1. Isolate the Absolute Value:
Start by subtracting 6 from both sides to isolate the absolute value:
[tex]\[
|x - 4| + 6 - 6 = 17 - 6
\][/tex]
Simplifying gives:
[tex]\[
|x - 4| = 11
\][/tex]
2. Solve the Absolute Value Equation:
The equation [tex]\( |x - 4| = 11 \)[/tex] means that the expression inside the absolute value can either be 11 or -11. So we have two cases to consider:
Case 1: [tex]\( x - 4 = 11 \)[/tex]
- Add 4 to both sides:
[tex]\[
x = 11 + 4
\][/tex]
- So, [tex]\( x = 15 \)[/tex].
Case 2: [tex]\( x - 4 = -11 \)[/tex]
- Add 4 to both sides:
[tex]\[
x = -11 + 4
\][/tex]
- So, [tex]\( x = -7 \)[/tex].
3. Solution:
The solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
Thus, the correct answer is option C: [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
1. Isolate the Absolute Value:
Start by subtracting 6 from both sides to isolate the absolute value:
[tex]\[
|x - 4| + 6 - 6 = 17 - 6
\][/tex]
Simplifying gives:
[tex]\[
|x - 4| = 11
\][/tex]
2. Solve the Absolute Value Equation:
The equation [tex]\( |x - 4| = 11 \)[/tex] means that the expression inside the absolute value can either be 11 or -11. So we have two cases to consider:
Case 1: [tex]\( x - 4 = 11 \)[/tex]
- Add 4 to both sides:
[tex]\[
x = 11 + 4
\][/tex]
- So, [tex]\( x = 15 \)[/tex].
Case 2: [tex]\( x - 4 = -11 \)[/tex]
- Add 4 to both sides:
[tex]\[
x = -11 + 4
\][/tex]
- So, [tex]\( x = -7 \)[/tex].
3. Solution:
The solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
Thus, the correct answer is option C: [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
