Answer :

To solve the equation [tex]\(4|x+1|-3=8\)[/tex], let's go through the steps together:

1. Start by isolating the absolute value expression.
Add 3 to both sides of the equation:
[tex]\[
4|x+1| = 11
\][/tex]

2. Divide both sides by 4 to solve for the absolute value.
[tex]\[
|x+1| = \frac{11}{4}
\][/tex]

3. Set up two separate equations based on the absolute value property.
The absolute value equation [tex]\( |x+1| = \frac{11}{4} \)[/tex] can be split into two equations:
[tex]\[
x + 1 = \frac{11}{4}
\][/tex]
[tex]\[
x + 1 = -\frac{11}{4}
\][/tex]

4. Solve each equation separately.

- First equation:
[tex]\[
x + 1 = \frac{11}{4}
\][/tex]
Subtract 1 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{11}{4} - 1 = \frac{11}{4} - \frac{4}{4} = \frac{7}{4}
\][/tex]

- Second equation:
[tex]\[
x + 1 = -\frac{11}{4}
\][/tex]
Subtract 1 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{11}{4} - 1 = -\frac{11}{4} - \frac{4}{4} = -\frac{15}{4}
\][/tex]

5. State the solutions.
The solutions for the equation [tex]\(4|x+1|-3=8\)[/tex] are:
[tex]\[
x = \frac{7}{4} \quad \text{and} \quad x = -\frac{15}{4}
\][/tex]

These are the values of [tex]\( x \)[/tex] that satisfy the original equation.