To find the value of [tex]\(\frac{6x}{5}\)[/tex] given the equation [tex]\(\frac{2}{x} - \frac{1}{3x} = 2\)[/tex], where [tex]\(x > 0\)[/tex], follow these detailed steps:
1. Simplify the equation:
[tex]\[
\frac{2}{x} - \frac{1}{3x} = 2
\][/tex]
Both terms on the left side have a common denominator of [tex]\(x\)[/tex]. Thus, you can combine them:
[tex]\[
\frac{2}{x} - \frac{1}{3x} = \frac{6}{3x} - \frac{1}{3x} = \frac{6 - 1}{3x} = \frac{5}{3x}
\][/tex]
So the equation becomes:
[tex]\[
\frac{5}{3x} = 2
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], multiply both sides of the equation by [tex]\(3x\)[/tex]:
[tex]\[
5 = 2 \cdot 3x
\][/tex]
Simplify:
[tex]\[
5 = 6x
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{5}{6}
\][/tex]
3. Find the value of [tex]\(\frac{6x}{5}\)[/tex]:
Substitute [tex]\(x = \frac{5}{6}\)[/tex] into [tex]\(\frac{6x}{5}\)[/tex]:
[tex]\[
\frac{6x}{5} = \frac{6 \left(\frac{5}{6}\right)}{5}
\][/tex]
Simplify the expression:
[tex]\[
\frac{6 \cdot \frac{5}{6}}{5} = \frac{5}{5} = 1
\][/tex]
Therefore, the value of [tex]\(\frac{6x}{5}\)[/tex] is [tex]\(\boxed{1}\)[/tex].
