Answer :
To solve the equation [tex]\( |x+5|-6=7 \)[/tex], we can take a step-by-step approach:
1. Isolate the Absolute Value:
Start by adding 6 to both sides of the equation to get:
[tex]\[
|x+5| = 13
\][/tex]
2. Consider the Two Cases of Absolute Value:
The equation [tex]\( |x+5| = 13 \)[/tex] can be split into two separate cases because absolute value measures the distance from zero:
- Case 1: [tex]\( x+5 = 13 \)[/tex]
- Case 2: [tex]\( x+5 = -13 \)[/tex]
3. Solve Case 1 ([tex]\( x+5 = 13 \)[/tex]):
Subtract 5 from both sides:
[tex]\[
x = 13 - 5 = 8
\][/tex]
4. Solve Case 2 ([tex]\( x+5 = -13 \)[/tex]):
Subtract 5 from both sides:
[tex]\[
x = -13 - 5 = -18
\][/tex]
5. Conclusion:
The solutions to the equation [tex]\( |x+5|-6=7 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = -18 \)[/tex].
Thus, the correct choice from the provided options is:
D. [tex]\( x=8 \)[/tex] and [tex]\( x=-18 \)[/tex].
1. Isolate the Absolute Value:
Start by adding 6 to both sides of the equation to get:
[tex]\[
|x+5| = 13
\][/tex]
2. Consider the Two Cases of Absolute Value:
The equation [tex]\( |x+5| = 13 \)[/tex] can be split into two separate cases because absolute value measures the distance from zero:
- Case 1: [tex]\( x+5 = 13 \)[/tex]
- Case 2: [tex]\( x+5 = -13 \)[/tex]
3. Solve Case 1 ([tex]\( x+5 = 13 \)[/tex]):
Subtract 5 from both sides:
[tex]\[
x = 13 - 5 = 8
\][/tex]
4. Solve Case 2 ([tex]\( x+5 = -13 \)[/tex]):
Subtract 5 from both sides:
[tex]\[
x = -13 - 5 = -18
\][/tex]
5. Conclusion:
The solutions to the equation [tex]\( |x+5|-6=7 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = -18 \)[/tex].
Thus, the correct choice from the provided options is:
D. [tex]\( x=8 \)[/tex] and [tex]\( x=-18 \)[/tex].