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Solve [tex]\( |x-5| + 7 = 13 \)[/tex].

A. [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex]

B. [tex]\( x = 11 \)[/tex] and [tex]\( x = -11 \)[/tex]

C. [tex]\( x = -11 \)[/tex] and [tex]\( x = 1 \)[/tex]

D. [tex]\( x = -11 \)[/tex] and [tex]\( x = -1 \)[/tex]

Sagot :

GinnyAnsweridn
To solve the equation [tex]\( |x - 5| + 7 = 13 \)[/tex], we will first isolate the absolute value term and then consider the two possible cases for the absolute value expression.

1. Start by isolating the absolute value term:
[tex]\[ |x - 5| + 7 = 13 \][/tex]
Subtract 7 from both sides:
[tex]\[ |x - 5| = 6 \][/tex]

2. Now, we need to solve the equation [tex]\( |x - 5| = 6 \)[/tex]. The absolute value equation [tex]\( |A| = B \)[/tex] implies two cases:
- [tex]\( A = B \)[/tex]
- [tex]\( A = -B \)[/tex]

Applying this to our equation [tex]\( |x - 5| = 6 \)[/tex], we get:
- [tex]\( x - 5 = 6 \)[/tex]
- [tex]\( x - 5 = -6 \)[/tex]

3. Solve each case separately:

Case 1: [tex]\( x - 5 = 6 \)[/tex]
[tex]\[ x - 5 = 6 \][/tex]
Add 5 to both sides:
[tex]\[ x = 11 \][/tex]

Case 2: [tex]\( x - 5 = -6 \)[/tex]
[tex]\[ x - 5 = -6 \][/tex]
Add 5 to both sides:
[tex]\[ x = -1 \][/tex]

4. Hence, the solutions to the equation [tex]\( |x - 5| + 7 = 13 \)[/tex] are [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex].

Therefore, the correct answer is:
A. [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex]
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