Sure, let's solve the given equation step by step.
Given equation:
[tex]\[
\frac{2}{3}x + x\left(\frac{1}{4} - \frac{2}{3}\right) = \frac{1}{2} \times \left(\frac{-5}{3}\right)
\][/tex]
Step 1: Simplify the equation. First, handle the right side:
[tex]\[
\frac{1}{2} \times \left(\frac{-5}{3}\right) = \frac{-5}{6}
\][/tex]
So, the equation now reads:
[tex]\[
\frac{2}{3}x + x\left(\frac{1}{4} - \frac{2}{3}\right) = \frac{-5}{6}
\][/tex]
Step 2: Simplify the expression inside the parenthesis:
[tex]\[
\frac{1}{4} - \frac{2}{3}
\][/tex]
To do this, find a common denominator, which is 12. Convert the fractions:
[tex]\[
\frac{1}{4} = \frac{3}{12}, \quad \frac{2}{3} = \frac{8}{12}
\][/tex]
Now subtract the fractions:
[tex]\[
\frac{3}{12} - \frac{8}{12} = \frac{-5}{12}
\][/tex]
So, the equation becomes:
[tex]\[
\frac{2}{3}x + x\left(\frac{-5}{12}\right) = \frac{-5}{6}
\][/tex]
Step 3: Combine like terms on the left side of the equation:
[tex]\[
\frac{2}{3}x - \frac{5}{12}x
\][/tex]
To combine these, again find a common denominator, which is 12:
[tex]\[
\frac{2}{3} = \frac{8}{12}
\][/tex]
So:
[tex]\[
\frac{8}{12}x - \frac{5}{12}x = \frac{3}{12}x
\][/tex]
Therefore, the equation simplifies to:
[tex]\[
\frac{3}{12}x = \frac{-5}{6}
\][/tex]
Simplify further by reducing the fraction:
[tex]\[
\frac{3}{12} = \frac{1}{4}
\][/tex]
So:
[tex]\[
\frac{1}{4}x = \frac{-5}{6}
\][/tex]
Step 4: Solve for [tex]\(x\)[/tex]. To isolate [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
x = \frac{-5}{6} \times 4
\][/tex]
Simplify the multiplication:
[tex]\[
x = \frac{-20}{6}
\][/tex]
Reduce the fraction:
[tex]\[
x = \frac{-10}{3}
\][/tex]
Thus, the solution to the equation is:
[tex]\[
x = -\frac{10}{3}
\][/tex]
This is the detailed, step-by-step solution.
