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Solve the equation [tex]|f-8|=54[/tex] for [tex]f[/tex].

A. [tex]f=62[/tex] and [tex]f=-46[/tex]

B. [tex]f=-64[/tex] and [tex]f=48[/tex]

C. [tex]f=448[/tex] and [tex]f=-448[/tex]

D. [tex]f=7[/tex] and [tex]f=-7[/tex]

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]

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