What is the solution to the equation [tex]\frac{1}{4} x - \frac{1}{8} = \frac{7}{8} + \frac{1}{2} x[/tex]?

A. [tex]x = -5[/tex]
B. [tex]x = -4[/tex]
C. [tex]x = 4[/tex]
D. [tex]x = 5[/tex]



Answer :

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]