To solve the equation [tex]\(\frac{1}{3} x + \frac{1}{4} = \frac{2}{5}\left(x - \frac{3}{4}\right)\)[/tex], follow these steps:
1. Expand the right-hand side (RHS) of the equation:
[tex]\[
\frac{1}{3} x + \frac{1}{4} = \frac{2}{5} x - \frac{2}{5} \cdot \frac{3}{4}
\][/tex]
Simplify the multiplication on the RHS:
[tex]\[
\frac{1}{3} x + \frac{1}{4} = \frac{2}{5} x - \frac{6}{20}
\][/tex]
Simplify [tex]\(\frac{6}{20}\)[/tex] to obtain:
[tex]\[
\frac{1}{3} x + \frac{1}{4} = \frac{2}{5} x - \frac{3}{10}
\][/tex]
2. Combine like terms to isolate [tex]\(x\)[/tex]:
Subtract [tex]\(\frac{1}{3} x\)[/tex] from both sides:
[tex]\[
\frac{1}{4} = \frac{2}{5} x - \frac{1}{3} x - \frac{3}{10}
\][/tex]
3. Combine the [tex]\(x\)[/tex] terms on the right-hand side:
Find a common denominator for [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]. The common denominator is 15:
[tex]\[
\frac{2}{5} x = \frac{6}{15} x
\][/tex]
[tex]\[
\frac{1}{3} x = \frac{5}{15} x
\][/tex]
Thus,
[tex]\[
\frac{6}{15} x - \frac{5}{15} x = \frac{1}{15} x
\][/tex]
The equation now looks like:
[tex]\[
\frac{1}{4} = \frac{1}{15} x - \frac{3}{10}
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
Add [tex]\(\frac{3}{10}\)[/tex] to both sides:
[tex]\[
\frac{1}{4} + \frac{3}{10} = \frac{1}{15} x
\][/tex]
Find a common denominator for [tex]\(\frac{1}{4}\)[/tex] and [tex]\(\frac{3}{10}\)[/tex]. The common denominator is 20:
[tex]\[
\frac{1}{4} = \frac{5}{20}
\][/tex]
[tex]\[
\frac{3}{10} = \frac{6}{20}
\][/tex]
Adding these together:
[tex]\[
\frac{5}{20} + \frac{6}{20} = \frac{11}{20}
\][/tex]
So,
[tex]\[
\frac{11}{20} = \frac{1}{15} x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Multiply both sides by 15:
[tex]\[
15 \cdot \frac{11}{20} = x
\][/tex]
Simplify the left-hand side:
[tex]\[
\frac{165}{20} = x
\][/tex]
Simplify [tex]\(\frac{165}{20}\)[/tex]:
[tex]\[
x = 8.25
\][/tex]
So, the solution to the equation is:
[tex]\[
x = 8.25
\][/tex]
