Find all solutions to the equation [tex]|4x - 5| = 7[/tex].

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

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

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

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

Answer :

To solve the equation
[tex]$$
|4x - 5| = 7,
$$[/tex]
we need to consider the two cases that arise from the definition of absolute value.

Case 1:
Assume the expression inside the absolute value is positive:
[tex]$$
4x - 5 = 7.
$$[/tex]
Add 5 to both sides:
[tex]$$
4x = 12.
$$[/tex]
Divide by 4:
[tex]$$
x = 3.
$$[/tex]

Case 2:
Assume the expression inside the absolute value is negative:
[tex]$$
4x - 5 = -7.
$$[/tex]
Add 5 to both sides:
[tex]$$
4x = -2.
$$[/tex]
Divide by 4:
[tex]$$
x = -0.5.
$$[/tex]

Thus, the solutions to the equation are
[tex]$$
x = 3 \quad \text{and} \quad x = -0.5.
$$[/tex]

Comparing with the provided options, the correct answer is:

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