Question

for what value of x do the expressions \\(\frac{2}{3}x + 2\\) and \\(\frac{4}{3}x - 6\\) have the same value?

Explanation:

Step1: Set the two expressions equal

To find the value of \( x \) where \( \frac{2}{3}x + 2 \) and \( \frac{4}{3}x - 6 \) are equal, we set up the equation:
\[
\frac{2}{3}x + 2 = \frac{4}{3}x - 6
\]

Step2: Subtract \( \frac{2}{3}x \) from both sides

Subtracting \( \frac{2}{3}x \) from each side to get the \( x \)-terms on one side:
\[
2 = \frac{4}{3}x - \frac{2}{3}x - 6
\]
Simplifying the right side, \( \frac{4}{3}x - \frac{2}{3}x = \frac{2}{3}x \), so the equation becomes:
\[
2 = \frac{2}{3}x - 6
\]

Step3: Add 6 to both sides

Adding 6 to both sides to isolate the term with \( x \):
\[
2 + 6 = \frac{2}{3}x
\]
Simplifying the left side:
\[
8 = \frac{2}{3}x
\]

Step4: Solve for \( x \)

Multiply both sides by \( \frac{3}{2} \) to solve for \( x \):
\[
x = 8 \times \frac{3}{2}
\]
Simplifying the right side, \( 8 \times \frac{3}{2} = 12 \), so:
\[
x = 12
\]

Answer:

\( 12 \)