To solve the equation [tex]\(\frac{5}{2} x - \left(\frac{2}{5} - \frac{1}{4} x\right) = \frac{5}{4} x + \frac{1}{10}\)[/tex], follow these steps:
1. Distribute and simplify the terms inside the parentheses:
The given equation is:
[tex]\[
\frac{5}{2} x - \left(\frac{2}{5} - \frac{1}{4} x\right) = \frac{5}{4} x + \frac{1}{10}
\][/tex]
Distribute the negative sign inside the parentheses:
[tex]\[
\frac{5}{2} x - \frac{2}{5} + \frac{1}{4} x = \frac{5}{4} x + \frac{1}{10}
\][/tex]
2. Combine like terms on the left-hand side (LHS):
On the left-hand side, combine [tex]\(\frac{5}{2} x\)[/tex] and [tex]\(\frac{1}{4} x\)[/tex]:
[tex]\[
\left(\frac{5}{2} + \frac{1}{4}\right) x - \frac{2}{5} = \frac{5}{4} x + \frac{1}{10}
\][/tex]
To combine [tex]\(\frac{5}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex], find a common denominator, which is 4:
[tex]\[
\frac{5}{2} = \frac{10}{4}
\][/tex]
So,
[tex]\[
\left(\frac{10}{4} + \frac{1}{4}\right) x - \frac{2}{5} = \frac{5}{4} x + \frac{1}{10}
\][/tex]
[tex]\[
\frac{11}{4} x - \frac{2}{5} = \frac{5}{4} x + \frac{1}{10}
\][/tex]
3. Isolate the variable x:
To isolate [tex]\(x\)[/tex], move [tex]\(\frac{5}{4} x\)[/tex] to the left-hand side and [tex]\(\frac{2}{5}\)[/tex] to the right-hand side of the equation:
[tex]\[
\frac{11}{4} x - \frac{5}{4} x = \frac{2}{5} + \frac{1}{10}
\][/tex]
4. Combine the [tex]\(x\)[/tex] terms on the left-hand side:
[tex]\[
\left(\frac{11}{4} - \frac{5}{4}\right) x = \frac{2}{5} + \frac{1}{10}
\][/tex]
[tex]\[
\frac{6}{4} x = \frac{2}{5} + \frac{1}{10}
\][/tex]
Simplify [tex]\(\frac{6}{4}\)[/tex]:
[tex]\[
\frac{3}{2} x = \frac{2}{5} + \frac{1}{10}
\][/tex]
5. Find a common denominator for the terms on the right-hand side:
The common denominator for [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{10}\)[/tex] is 10:
[tex]\[
\frac{2}{5} = \frac{4}{10}
\][/tex]
So,
[tex]\[
\frac{3}{2} x = \frac{4}{10} + \frac{1}{10}
\][/tex]
[tex]\[
\frac{3}{2} x = \frac{5}{10}
\][/tex]
[tex]\[
\frac{3}{2} x = \frac{1}{2}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{3}{2}\)[/tex], which is [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
x = \frac{1}{2} \times \frac{2}{3}
\][/tex]
[tex]\[
x = \frac{2}{6}
\][/tex]
Simplify [tex]\(\frac{2}{6}\)[/tex]:
[tex]\[
x = \frac{1}{3}
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = \frac{1}{3} = 0.333333333333333
\][/tex]
