To solve the equation [tex]\( x - \frac{7}{8} = 4x + \frac{1}{2} \)[/tex], we will follow these steps:
1. Isolate the variable on one side:
Start by moving the terms involving [tex]\( x \)[/tex] to one side of the equation and the constant terms to the opposite side. Let's subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[
x - \frac{7}{8} - 4x = \frac{1}{2}
\][/tex]
Simplify the left side:
[tex]\[
-3x - \frac{7}{8} = \frac{1}{2}
\][/tex]
2. Combine like terms involving constants:
Add [tex]\( \frac{7}{8} \)[/tex] to both sides to isolate the terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-3x = \frac{1}{2} + \frac{7}{8}
\][/tex]
3. Find a common denominator to combine fractions:
The common denominator for [tex]\( \frac{1}{2} \)[/tex] and [tex]\( \frac{7}{8} \)[/tex] is 8. Therefore:
[tex]\[
\frac{1}{2} = \frac{4}{8}
\][/tex]
Now, add these fractions:
[tex]\[
\frac{4}{8} + \frac{7}{8} = \frac{11}{8}
\][/tex]
So the equation becomes:
[tex]\[
-3x = \frac{11}{8}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by -3 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\frac{11}{8}}{-3}
\][/tex]
Simplifying this we get:
[tex]\[
x = \frac{11}{8} \cdot \frac{-1}{3} = \frac{-11}{24}
\][/tex]
Therefore, the solution to the equation [tex]\( x - \frac{7}{8} = 4x + \frac{1}{2} \)[/tex] is:
[tex]\[
x = -\frac{11}{24}
\][/tex]
