Answer :
To solve the equation [tex]\(|x+5| - 6 = 7\)[/tex], we need to isolate the absolute value expression. Here are the steps:
1. Add 6 to both sides of the equation to eliminate the -6:
[tex]\[
|x+5| - 6 + 6 = 7 + 6
\][/tex]
Simplifying gives:
[tex]\[
|x+5| = 13
\][/tex]
2. Consider the definition of absolute value, which states that [tex]\(|a| = b\)[/tex] means [tex]\(a = b\)[/tex] or [tex]\(a = -b\)[/tex]. Here, [tex]\(a = x+5\)[/tex] and [tex]\(b = 13\)[/tex]. So, we have two possible equations:
- [tex]\(x + 5 = 13\)[/tex]
- [tex]\(x + 5 = -13\)[/tex]
3. Solve each equation separately:
- For [tex]\(x + 5 = 13\)[/tex], subtract 5 from both sides:
[tex]\[
x = 13 - 5
\][/tex]
Simplifying gives:
[tex]\[
x = 8
\][/tex]
- For [tex]\(x + 5 = -13\)[/tex], subtract 5 from both sides:
[tex]\[
x = -13 - 5
\][/tex]
Simplifying gives:
[tex]\[
x = -18
\][/tex]
4. Conclusion: The solutions to the equation [tex]\(|x+5| - 6 = 7\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = -18\)[/tex].
Therefore, the correct choice is [tex]\( \boxed{x = 8 \text{ and } x = -18} \)[/tex].
1. Add 6 to both sides of the equation to eliminate the -6:
[tex]\[
|x+5| - 6 + 6 = 7 + 6
\][/tex]
Simplifying gives:
[tex]\[
|x+5| = 13
\][/tex]
2. Consider the definition of absolute value, which states that [tex]\(|a| = b\)[/tex] means [tex]\(a = b\)[/tex] or [tex]\(a = -b\)[/tex]. Here, [tex]\(a = x+5\)[/tex] and [tex]\(b = 13\)[/tex]. So, we have two possible equations:
- [tex]\(x + 5 = 13\)[/tex]
- [tex]\(x + 5 = -13\)[/tex]
3. Solve each equation separately:
- For [tex]\(x + 5 = 13\)[/tex], subtract 5 from both sides:
[tex]\[
x = 13 - 5
\][/tex]
Simplifying gives:
[tex]\[
x = 8
\][/tex]
- For [tex]\(x + 5 = -13\)[/tex], subtract 5 from both sides:
[tex]\[
x = -13 - 5
\][/tex]
Simplifying gives:
[tex]\[
x = -18
\][/tex]
4. Conclusion: The solutions to the equation [tex]\(|x+5| - 6 = 7\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = -18\)[/tex].
Therefore, the correct choice is [tex]\( \boxed{x = 8 \text{ and } x = -18} \)[/tex].