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=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 work through each of the given questions step by step.

### Question 1:

We are given the function:

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

We need to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex]. So we set up the equation:

[tex]\[ 4|x-5| + 3 = 15 \][/tex]

1. Subtract 3 from both sides:

[tex]\[ 4|x-5| = 12 \][/tex]

2. Divide both sides by 4:

[tex]\[ |x-5| = 3 \][/tex]

3. Solve the absolute value equation. This means we solve the two equations:

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

4. Solve each equation:

- [tex]\( x - 5 = 3 \)[/tex] → [tex]\( x = 8 \)[/tex]
- [tex]\( x - 5 = -3 \)[/tex] → [tex]\( x = 2 \)[/tex]

So, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].

### Question 2:

We are given the function:

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

We need to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 6 \)[/tex]. So we set up the equation:

[tex]\[ -0.5|2x + 2| + 1 = 6 \][/tex]

1. Subtract 1 from both sides:

[tex]\[ -0.5|2x + 2| = 5 \][/tex]

2. Divide both sides by -0.5:

[tex]\[ |2x + 2| = -10 \][/tex]

An absolute value cannot equal a negative number. Therefore, there is no solution for [tex]\( x \)[/tex] that satisfies the equation [tex]\( f(x) = 6 \)[/tex].

So, for the second function, the solution is "no solution."