Certainly! Let's solve the equation step-by-step:
[tex]\[ \frac{2}{3} x + x \left( \frac{1}{4} - \frac{2}{3} \right) = \frac{1}{2} \times \frac{-5}{3} \][/tex]
1. Simplify the equation inside the parentheses:
[tex]\[ \frac{1}{4} - \frac{2}{3} \][/tex]
To subtract these fractions, we need a common denominator. The common denominator for 4 and 3 is 12. Convert both fractions:
[tex]\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \][/tex]
[tex]\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \][/tex]
Now perform the subtraction:
[tex]\[ \frac{3}{12} - \frac{8}{12} = -\frac{5}{12} \][/tex]
So the equation becomes:
[tex]\[ \frac{2}{3} x + x \left( -\frac{5}{12} \right) = \frac{1}{2} \times \frac{-5}{3} \][/tex]
2. Distribute the [tex]\( x \)[/tex] in the parentheses:
[tex]\[ \frac{2}{3} x + x \left( -\frac{5}{12} \right) = \frac{2}{3} x - \frac{5}{12} x \][/tex]
3. Combine like terms:
To combine [tex]\(\frac{2}{3} x\)[/tex] and [tex]\(- \frac{5}{12} x\)[/tex], we need a common denominator. The common denominator for 3 and 12 is 12. Convert both fractions:
[tex]\[ \frac{2}{3} x = \frac{2 \times 4}{3 \times 4} x = \frac{8}{12} x \][/tex]
So:
[tex]\[ \frac{8}{12} x - \frac{5}{12} x = \left( \frac{8}{12} - \frac{5}{12} \right) x = \frac{3}{12} x = \frac{1}{4} x \][/tex]
Now the equation is:
[tex]\[ \frac{1}{4} x = \frac{1}{2} \times \frac{-5}{3} \][/tex]
4. Simplify the right-hand side:
[tex]\[ \frac{1}{2} \times \frac{-5}{3} = \frac{1 \times (-5)}{2 \times 3} = \frac{-5}{6} \][/tex]
So the equation now is:
[tex]\[ \frac{1}{4} x = \frac{-5}{6} \][/tex]
5. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], multiply both sides of the equation by the reciprocal of [tex]\(\frac{1}{4}\)[/tex], which is 4:
[tex]\[ x = 4 \times \frac{-5}{6} \][/tex]
Simplify the right-hand side:
[tex]\[ x = \frac{4 \times (-5)}{6} = \frac{-20}{6} = -\frac{10}{3} \][/tex]
Thus, the solution to the equation is:
[tex]\[ x = -\frac{10}{3} \][/tex]
