Solving the Equation for the Value of x

The problem asks us to determine the value of the variable x given a specific algebraic equation involving fractions. The equation provided is:

\[\frac{5x}{2} - \frac{1}{4} \left( 6x - \frac{5}{3} \right) = \frac{7}{6}\]

We will solve this equation step-by-step to find the precise value of x.

Step 1: Distribute the Outer Fraction

The first step is to simplify the equation by distributing the fraction \(-\frac{1}{4}\) to both terms inside the parentheses \(\left( 6x - \frac{5}{3} \right)\).

\[\frac{5x}{2} - \left( \frac{1}{4} \times 6x \right) - \left( \frac{1}{4} \times -\frac{5}{3} \right) = \frac{7}{6}\]

Performing the multiplication gives:

\[\frac{5x}{2} - \frac{6x}{4} + \frac{5}{12} = \frac{7}{6}\]

Step 2: Simplify Fractional Coefficients

We can simplify the term \(\frac{6x}{4}\). Both the numerator and the denominator are divisible by 2.

\[\frac{6x}{4} = \frac{3x}{2}\]

Substitute this simplified term back into the equation:

\[\frac{5x}{2} - \frac{3x}{2} + \frac{5}{12} = \frac{7}{6}\]

Step 3: Combine Like Terms (x-terms)

Now, combine the terms that contain x, which are \(\frac{5x}{2}\) and \(-\frac{3x}{2}\). Since they have a common denominator, we can subtract their numerators directly.

\[\left( \frac{5x}{2} - \frac{3x}{2} \right) + \frac{5}{12} = \frac{7}{6}\]

\[\frac{5x - 3x}{2} + \frac{5}{12} = \frac{7}{6}\]

\[\frac{2x}{2} + \frac{5}{12} = \frac{7}{6}\]

Simplify \(\frac{2x}{2}\) to x:

\[x + \frac{5}{12} = \frac{7}{6}\]

Step 4: Isolate the Variable x

To find the value of x, we need to get it by itself on one side of the equation. We can do this by subtracting \(\frac{5}{12}\) from both sides of the equation.

\[x = \frac{7}{6} - \frac{5}{12}\]

Step 5: Perform Fraction Subtraction

To subtract the fractions \(\frac{7}{6}\) and \(\frac{5}{12}\), we need a common denominator. The least common denominator for 6 and 12 is 12.

Convert \(\frac{7}{6}\) to an equivalent fraction with a denominator of 12:

\[\frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12}\]

Now substitute this back into the equation for x:

\[x = \frac{14}{12} - \frac{5}{12}\]

Subtract the numerators:

\[x = \frac{14 - 5}{12}\]

\[x = \frac{9}{12}\]

Step 6: Simplify the Final Result

The final step is to simplify the fraction \(\frac{9}{12}\). The greatest common divisor of 9 and 12 is 3. Divide both the numerator and the denominator by 3.

\[x = \frac{9 \div 3}{12 \div 3}\]

\[x = \frac{3}{4}\]

Final Value of x

The calculation shows that the value of x satisfying the given equation is \(\frac{3}{4}\).