Solve the equations and check your answer in the case:
`(2)/(3)(3x-2)=(4)/(5)(2x-3) -(4)/(3)`

Answer

To solve the equation \(\frac{2}{3}(3x-2) = \frac{4}{5}(2x-3) - \frac{4}{3}\), we will follow these steps:

### Step 1: Distribute the fractions on both sides of the equation.
We start by distributing \(\frac{2}{3}\) on the left side and \(\frac{4}{5}\) on the right side.

\[ \frac{2}{3}(3x) - \frac{2}{3}(2) = \frac{4}{5}(2x) - \frac{4}{5}(3) - \frac{4}{3} \]

This simplifies to:

\[ 2x - \frac{4}{3} = \frac{8}{5}x - \frac{12}{5} - \frac{4}{3} \]

### Step 2: Find a common denominator for the fractions.
The common denominator for 5 and 3 is 15. We will convert all fractions to have a denominator of 15.

\[ 2x - \frac{4 \times 5}{15} = \frac{8 \times 3}{15}x - \frac{12 \times 3}{15} - \frac{4 \times 5}{15} \]

This gives us:

\[ 2x - \frac{20}{15} = \frac{24}{15}x - \frac{36}{15} - \frac{20}{15} \]

### Step 3: Combine like terms on the right side.
Combine the constants on the right side:

\[ 2x - \frac{20}{15} = \frac{24}{15}x - \frac{56}{15} \]

### Step 4: Move all terms involving \(x\) to one side and constant terms to the other side.
Rearranging gives:

\[ 2x - \frac{24}{15}x = -\frac{56}{15} + \frac{20}{15} \]

This simplifies to:

\[ 2x - \frac{24}{15}x = -\frac{36}{15} \]

### Step 5: Convert \(2x\) to have a common denominator.
Convert \(2x\) to have a denominator of 15:

\[ \frac{30}{15}x - \frac{24}{15}x = -\frac{36}{15} \]

### Step 6: Combine the \(x\) terms.
This gives:

\[ \frac{6}{15}x = -\frac{36}{15} \]

### Step 7: Solve for \(x\).
Multiplying both sides by \(\frac{15}{6}\):

\[ x = -\frac{36}{6} = -6 \]

### Step 8: Check the solution.
Substituting \(x = -6\) back into the original equation:

Left side:

\[ \frac{2}{3}(3(-6) - 2) = \frac{2}{3}(-18 - 2) = \frac{2}{3}(-20) = -\frac{40}{3} \]

Right side:

\[ \frac{4}{5}(2(-6) - 3) - \frac{4}{3} = \frac{4}{5}(-12 - 3) - \frac{4}{3} = \frac{4}{5}(-15) - \frac{4}{3} = -12 - \frac{4}{3} \]

Finding a common denominator for \(-12\) and \(-\frac{4}{3}\):

\[ -12 = -\frac{36}{3} \implies -\frac{36}{3} - \frac{4}{3} = -\frac{40}{3} \]

Both sides equal \(-\frac{40}{3}\), confirming our solution is correct.

### Final Answer:
\[ x = -6 \]