Let's solve the equation step-by-step:
Given:
[tex]\[
\frac{1}{x} + \frac{1}{2x} = -1
\][/tex]
Step 1: Find a common denominator for the terms on the left side of the equation.
The common denominator between [tex]\(x\)[/tex] and [tex]\(2x\)[/tex] is [tex]\(2x\)[/tex]. Rewrite each term using this common denominator:
[tex]\[
\frac{2}{2x} + \frac{1}{2x} = -1
\][/tex]
Step 2: Combine the fractions on the left side:
[tex]\[
\frac{2 + 1}{2x} = -1
\][/tex]
[tex]\[
\frac{3}{2x} = -1
\][/tex]
Step 3: Eliminate the fraction by multiplying both sides of the equation by [tex]\(2x\)[/tex]:
[tex]\[
3 = -2x
\][/tex]
Step 4: Solve for [tex]\(x\)[/tex] by isolating [tex]\(x\)[/tex]. Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
x = \frac{3}{-2}
\][/tex]
[tex]\[
x = -\frac{3}{2}
\][/tex]
So, the solution is:
[tex]\[
x = -\frac{3}{2}
\][/tex]
This is the value of [tex]\(x\)[/tex] that satisfies the given equation.
