To solve the equation [tex]\( |x-4| + 6 = 15 \)[/tex], follow these steps:
1. Isolate the Absolute Value:
Subtract 6 from both sides of the equation to isolate the absolute value term:
[tex]\[
|x-4| + 6 - 6 = 15 - 6
\][/tex]
[tex]\[
|x-4| = 9
\][/tex]
2. Consider the Definition of Absolute Value:
The equation [tex]\( |x-4| = 9 \)[/tex] means that [tex]\( x-4 \)[/tex] can be either 9 or -9. This gives us two separate equations to solve:
[tex]\[
x-4 = 9
\][/tex]
[tex]\[
x-4 = -9
\][/tex]
3. Solve Each Equation:
- For [tex]\( x-4 = 9 \)[/tex]:
[tex]\[
x = 9 + 4
\][/tex]
[tex]\[
x = 13
\][/tex]
- For [tex]\( x-4 = -9 \)[/tex]:
[tex]\[
x = -9 + 4
\][/tex]
[tex]\[
x = -5
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\( |x-4| + 6 = 15 \)[/tex] are:
[tex]\[
x = 13 \quad \text{and} \quad x = -5
\][/tex]
Therefore, the correct answer is:
C. [tex]\( x = 13 \)[/tex] and [tex]\( x = -5 \)[/tex]
