To solve the equation [tex]\(\frac{3}{2} = \frac{3}{2x} - \frac{6}{5x}\)[/tex], follow these steps:
1. Combine Fractions on the Right-Hand Side:
To combine the fractions on the right-hand side, find a common denominator. The denominators are [tex]\(2x\)[/tex] and [tex]\(5x\)[/tex]. The least common multiple (LCM) of these denominators is [tex]\(10x\)[/tex].
Rewrite each fraction with the common denominator [tex]\(10x\)[/tex]:
[tex]\[
\frac{3}{2x} = \frac{3 \cdot 5}{2x \cdot 5} = \frac{15}{10x}
\][/tex]
[tex]\[
\frac{6}{5x} = \frac{6 \cdot 2}{5x \cdot 2} = \frac{12}{10x}
\][/tex]
2. Rewrite the Right-Hand Side:
Substitute these fractions back into the equation:
[tex]\[
\frac{3}{2} = \frac{15}{10x} - \frac{12}{10x}
\][/tex]
Combine the fractions on the right-hand side:
[tex]\[
\frac{3}{2} = \frac{15 - 12}{10x} = \frac{3}{10x}
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
Now the equation is simplified to:
[tex]\[
\frac{3}{2} = \frac{3}{10x}
\][/tex]
Remove the common factor of 3 from both sides:
[tex]\[
\frac{1}{2} = \frac{1}{10x}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
10x \cdot \frac{1}{2} = 1
\][/tex]
Simplify the equation:
[tex]\[
5x = 1
\][/tex]
Divide both sides by 5:
[tex]\[
x = \frac{1}{5}
\][/tex]
5. Conclusion:
Only one solution exists, and it is:
[tex]\[
x = \frac{1}{5}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x = \frac{1}{5}}
\][/tex]
