Solve [tex]|x-5| + 7 = 17[/tex]

A. [tex]x = 15[/tex] and [tex]x = -15[/tex]
B. [tex]x = -15[/tex] and [tex]x = 5[/tex]
C. [tex]x = 15[/tex] and [tex]x = -5[/tex]
D. [tex]x = -15[/tex] and [tex]x = -5[/tex]



Answer :

To solve the equation [tex]\( |x - 5| + 7 = 17 \)[/tex], follow these steps:

1. Isolate the absolute value expression:

Start with the given equation:
[tex]\[ |x - 5| + 7 = 17 \][/tex]

Subtract 7 from both sides to isolate the absolute value term:
[tex]\[ |x - 5| = 17 - 7 \][/tex]
Simplifying this gives:
[tex]\[ |x - 5| = 10 \][/tex]

2. Break down the absolute value equation into two cases:

The absolute value equation [tex]\( |x - 5| = 10 \)[/tex] implies two scenarios:

- Case 1: [tex]\( x - 5 = 10 \)[/tex]
- Case 2: [tex]\( x - 5 = -10 \)[/tex]

3. Solve each case separately:

- For Case 1:
[tex]\[ x - 5 = 10 \][/tex]
Add 5 to both sides:
[tex]\[ x = 10 + 5 \][/tex]
[tex]\[ x = 15 \][/tex]

- For Case 2:
[tex]\[ x - 5 = -10 \][/tex]
Add 5 to both sides:
[tex]\[ x = -10 + 5 \][/tex]
[tex]\[ x = -5 \][/tex]

4. Conclusion:

The solutions to the equation [tex]\( |x - 5| + 7 = 17 \)[/tex] are [tex]\( x = 15 \)[/tex] and [tex]\( x = -5 \)[/tex].

Therefore, the correct option is:
[tex]\[ \boxed{C. \, x=15 \text{ and } x=-5} \][/tex]