To solve the equation [tex]\(\sqrt[3]{\left(\frac{1}{8} - x\right)} = -\frac{1}{2}\)[/tex], follow these steps:
1. Cube both sides to eliminate the cube root:
[tex]\[
\left(\sqrt[3]{\left(\frac{1}{8} - x\right)}\right)^3 = \left(-\frac{1}{2}\right)^3
\][/tex]
This simplifies to:
[tex]\[
\frac{1}{8} - x = -\frac{1}{8}
\][/tex]
2. Isolate [tex]\( x \)[/tex] by adding [tex]\( \frac{1}{8} \)[/tex] to both sides:
[tex]\[
\frac{1}{8} - x + x = -\frac{1}{8} + x + x
\][/tex]
Therefore:
[tex]\[
\frac{1}{8} = -\frac{1}{8} + x
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{1}{8} + \frac{1}{8}
\][/tex]
Adding the fractions:
[tex]\[
x = \frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4}
\][/tex]
The solution is [tex]\( x = \boxed{\frac{1}{4}} \)[/tex].
