Answer :
To solve the equation [tex]\( |f - 8| = 54 \)[/tex], we need to consider the definition of absolute value. The absolute value equation [tex]\( |A| = B \)[/tex] means that [tex]\( A \)[/tex] can be either [tex]\( B \)[/tex] or [tex]\(-B\)[/tex]. Applying this to our problem:
### Step-by-Step Solution:
1. Set up the two possible equations based on the property of absolute values:
[tex]\[
f - 8 = 54 \quad \text{or} \quad f - 8 = -54
\][/tex]
2. Solve the first equation [tex]\( f - 8 = 54 \)[/tex]:
[tex]\[
f - 8 = 54
\][/tex]
Add 8 to both sides:
[tex]\[
f = 54 + 8
\][/tex]
[tex]\[
f = 62
\][/tex]
3. Solve the second equation [tex]\( f - 8 = -54 \)[/tex]:
[tex]\[
f - 8 = -54
\][/tex]
Add 8 to both sides:
[tex]\[
f = -54 + 8
\][/tex]
[tex]\[
f = -46
\][/tex]
4. Combine the solutions:
The solutions to the equation [tex]\( |f - 8| = 54 \)[/tex] are
[tex]\[
f = 62 \quad \text{and} \quad f = -46
\][/tex]
Therefore, the correct answer is:
[tex]\[
f = 62 \quad \text{and} \quad f = -46
\][/tex]
### Step-by-Step Solution:
1. Set up the two possible equations based on the property of absolute values:
[tex]\[
f - 8 = 54 \quad \text{or} \quad f - 8 = -54
\][/tex]
2. Solve the first equation [tex]\( f - 8 = 54 \)[/tex]:
[tex]\[
f - 8 = 54
\][/tex]
Add 8 to both sides:
[tex]\[
f = 54 + 8
\][/tex]
[tex]\[
f = 62
\][/tex]
3. Solve the second equation [tex]\( f - 8 = -54 \)[/tex]:
[tex]\[
f - 8 = -54
\][/tex]
Add 8 to both sides:
[tex]\[
f = -54 + 8
\][/tex]
[tex]\[
f = -46
\][/tex]
4. Combine the solutions:
The solutions to the equation [tex]\( |f - 8| = 54 \)[/tex] are
[tex]\[
f = 62 \quad \text{and} \quad f = -46
\][/tex]
Therefore, the correct answer is:
[tex]\[
f = 62 \quad \text{and} \quad f = -46
\][/tex]
