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

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: [ 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 )