College

Find the solution for [tex]$x$[/tex] if [tex]$|4x - 5| = 7$[/tex].

A. [tex]$x = -3$[/tex] and [tex]$x = -0.5$[/tex]

B. [tex]$x = -3$[/tex] and [tex]$x = 0.5$[/tex]

C. [tex]$x = -0.5$[/tex] and [tex]$x = 3$[/tex]

D. [tex]$x = 0.5$[/tex] and [tex]$x = 3$[/tex]

Answer :

To solve the equation [tex]\( |4x - 5| = 7 \)[/tex], we need to consider two separate cases because the absolute value can represent a positive or a negative situation. Here are the steps:

1. Case 1: Positive Situation
- Start with the equation without the absolute value:
[tex]\[
4x - 5 = 7
\][/tex]
- Add 5 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
4x = 12
\][/tex]
- Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{12}{4} = 3
\][/tex]

2. Case 2: Negative Situation
- Set the expression inside the absolute value equal to the negative of 7:
[tex]\[
4x - 5 = -7
\][/tex]
- Add 5 to both sides:
[tex]\[
4x = -2
\][/tex]
- Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-2}{4} = -0.5
\][/tex]

The solutions are [tex]\( x = 3 \)[/tex] and [tex]\( x = -0.5 \)[/tex]. Therefore, the correct option from those provided is:

c.) [tex]\( x = -0.5 \)[/tex] and [tex]\( x = 3 \)[/tex]