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Given the function [tex]$f(x) = 0.5|x-4| - 3$[/tex], for what values of [tex]$x$[/tex] is [tex]$f(x) = 7$[/tex]?

A. [tex]$x = -24, x = 16$[/tex]
B. [tex][tex]$x = -16, x = 24$[/tex][/tex]
C. [tex]$x = -1, x = 9$[/tex]
D. [tex]$x = 1, x = -9$[/tex]

Sagot :

GinnyAnswer
Let's solve the given equation [tex]\( f(x) = 7 \)[/tex] for the function [tex]\( f(x) = 0.5|x-4|-3 \)[/tex].

1. Start with the equation:
[tex]\[ f(x) = 7 \][/tex]
Substitute [tex]\( f(x) \)[/tex] with [tex]\( 0.5|x-4|-3 \)[/tex]:
[tex]\[ 0.5|x-4| - 3 = 7 \][/tex]

2. Add 3 to both sides of the equation to isolate the absolute value term:
[tex]\[ 0.5|x-4| = 10 \][/tex]

3. Multiply both sides by 2 to further isolate the absolute value term:
[tex]\[ |x-4| = 20 \][/tex]

4. The equation [tex]\( |x-4| = 20 \)[/tex] gives us two possible cases:
[tex]\[ x - 4 = 20 \quad \text{or} \quad x - 4 = -20 \][/tex]

5. Solve each case separately:
- Case 1: [tex]\( x - 4 = 20 \)[/tex]
[tex]\[ x = 20 + 4 \][/tex]
[tex]\[ x = 24 \][/tex]

- Case 2: [tex]\( x - 4 = -20 \)[/tex]
[tex]\[ x = -20 + 4 \][/tex]
[tex]\[ x = -16 \][/tex]

Thus, the solutions for the equation [tex]\( f(x) = 7 \)[/tex] are [tex]\( x = 24 \)[/tex] and [tex]\( x = -16 \)[/tex].

The correct answer is:
[tex]\[ x = -16, x = 24 \][/tex]
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