College

Given the function [tex]f(x)=4|x-5|+3[/tex], for what values of [tex]x[/tex] is [tex]f(x)=15[/tex]?

A. [tex]x=0.2, x=8[/tex]
B. [tex]x=1.5, x=8[/tex]
C. [tex]x=2, x=7.5[/tex]
D. [tex]x=0.5, x=7.5[/tex]

Given the function [tex]f(x)=-0.5|2x+2|+1[/tex], for what values of [tex]x[/tex] is [tex]f(x)=6[/tex]?

A. [tex]x=6, x=-5[/tex]
B. [tex]x=5, x=-5[/tex]
C. [tex]x=7, x=-6[/tex]
D. No solution

Answer :

Let's solve each part of the question step by step.

### First Function: [tex]\( f(x) = 4|x-5| + 3 \)[/tex]

We need to find values of [tex]\( x \)[/tex] such that [tex]\( f(x) = 15 \)[/tex].

1. Set the function equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]

2. Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x-5| = 12
\][/tex]

3. Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]

4. Solve the absolute value equation. An absolute value equation [tex]\( |a| = b \)[/tex] has two solutions: [tex]\( a = b \)[/tex] and [tex]\( a = -b \)[/tex].

- Case 1: [tex]\( x-5 = 3 \)[/tex]
[tex]\[
x = 8
\][/tex]

- Case 2: [tex]\( x-5 = -3 \)[/tex]
[tex]\[
x = 2
\][/tex]

Therefore, the values of [tex]\( x \)[/tex] that satisfy the equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].

### Second Function: [tex]\( f(x) = -0.5|2x+2| + 1 \)[/tex]

We need to find values of [tex]\( x \)[/tex] such that [tex]\( f(x) = 6 \)[/tex].

1. Set the function equal to 6:
[tex]\[
-0.5|2x+2| + 1 = 6
\][/tex]

2. Subtract 1 from both sides:
[tex]\[
-0.5|2x+2| = 5
\][/tex]

3. Divide by -0.5, which will flip the equation since we are dividing by a negative number:
[tex]\[
|2x+2| = -10
\][/tex]

The absolute value [tex]\( |2x+2| = -10 \)[/tex] does not have any solutions because an absolute value cannot be negative. Hence, there is no solution for [tex]\( x \)[/tex].

In summary:

- For the first function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], the solutions are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].

- For the second function [tex]\( f(x) = -0.5|2x+2| + 1 \)[/tex], there is no solution.