Let's solve the equation step-by-step:
The given equation is:
[tex]\[
\frac{1}{4} x - \frac{1}{8} = \frac{7}{8} + \frac{1}{2} x
\][/tex]
1. Combine like terms involving [tex]\( x \)[/tex] on one side of the equation:
First, move [tex]\(\frac{1}{2} x\)[/tex] from the right side to the left side by subtracting [tex]\(\frac{1}{2} x\)[/tex] from both sides:
[tex]\[
\frac{1}{4} x - \frac{1}{2} x - \frac{1}{8} = \frac{7}{8}
\][/tex]
2. Simplify the terms involving [tex]\( x \)[/tex]:
Convert [tex]\(\frac{1}{2} x\)[/tex] to a fraction with the same denominator as [tex]\(\frac{1}{4} x\)[/tex]:
[tex]\[
\frac{1}{2} x = \frac{2}{4} x
\][/tex]
So, the equation becomes:
[tex]\[
\frac{1}{4} x - \frac{2}{4} x - \frac{1}{8} = \frac{7}{8}
\][/tex]
3. Combine the [tex]\( x \)[/tex] terms:
[tex]\[
\left( \frac{1}{4} - \frac{2}{4} \right) x - \frac{1}{8} = \frac{7}{8}
\][/tex]
Simplify:
[tex]\[
-\frac{1}{4} x - \frac{1}{8} = \frac{7}{8}
\][/tex]
4. Isolate the term with [tex]\( x \)[/tex]:
Add [tex]\(\frac{1}{8}\)[/tex] to both sides to move the constant term to the right:
[tex]\[
-\frac{1}{4} x = \frac{7}{8} + \frac{1}{8}
\][/tex]
Combine the constants on the right side:
[tex]\[
-\frac{1}{4} x = \frac{8}{8} = 1
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], multiply both sides by [tex]\(-4\)[/tex]:
[tex]\[
x = -4
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{1}{4} x - \frac{1}{8} = \frac{7}{8} + \frac{1}{2} x\)[/tex] is:
[tex]\[
x = -4
\][/tex]
Among the given options, the correct answer is:
[tex]\[
x = -4
\][/tex]
