High School

What are the solutions to [tex]-4|x-1|+3=-1[/tex]?

A. [tex]x = 2[/tex]

B. [tex]x = -\frac{7}{4}[/tex] and [tex]x = \frac{7}{4}[/tex]

C. [tex]x = -2[/tex] and [tex]x = 0[/tex]

D. [tex]x = \frac{9}{4}[/tex] and [tex]x = \frac{7}{4}[/tex]

E. [tex]x = 0[/tex] and [tex]x = 2[/tex]

Answer :

To solve the equation [tex]\(-4|x-1| + 3 = -1\)[/tex], let's follow these steps:

1. Isolate the Absolute Value:
Start by moving the constant on the left side to the right side of the equation:
[tex]\[
-4|x-1| + 3 = -1
\][/tex]
Subtract 3 from both sides:
[tex]\[
-4|x-1| = -1 - 3
\][/tex]
[tex]\[
-4|x-1| = -4
\][/tex]

2. Divide by [tex]\(-4\)[/tex] to Simplify:
Divide both sides of the equation by [tex]\(-4\)[/tex] to solve for the absolute value:
[tex]\[
|x-1| = \frac{-4}{-4}
\][/tex]
[tex]\[
|x-1| = 1
\][/tex]

3. Set Up the Two Cases for the Absolute Value:
The absolute value equation [tex]\(|x-1| = 1\)[/tex] gives us two possible scenarios:

- Case 1: [tex]\(x - 1 = 1\)[/tex]
- Case 2: [tex]\(x - 1 = -1\)[/tex]

4. Solve Each Case:

- Case 1:
[tex]\[
x - 1 = 1
\][/tex]
Add 1 to both sides:
[tex]\[
x = 1 + 1 = 2
\][/tex]

- Case 2:
[tex]\[
x - 1 = -1
\][/tex]
Add 1 to both sides:
[tex]\[
x = -1 + 1 = 0
\][/tex]

5. Solutions:
The solutions to the equation are [tex]\(x = 2\)[/tex] and [tex]\(x = 0\)[/tex].

Thus, the correct answer is:
- [tex]\(x = 2\)[/tex] and [tex]\(x = 0\)[/tex], which corresponds to option E.