Question - Solving a Fractional Inequality Problem

The provided inequality is:

\(\frac{4}{x-2} > \frac{7}{x-3}\)

To solve it, first find a common denominator:

\((x-2)(x-3)\)

Then multiply both sides by the common denominator, being careful to consider that it can change sign based on the value of \(x\):

\(4(x-3) > 7(x-2)\)

Now distribute and move all terms to one side:

\(4x - 12 > 7x - 14\)

Subtract \(4x\) and add \(14\) to both sides:

\(-12 + 14 > 7x - 4x\)

Simplify:

\(2 > 3x\)

Now, divide by 3:

\(\frac{2}{3} > x\)

So, the solution in terms of \(x\) is:

\(x < \frac{2}{3}\)

However, since the denominators in the original fractions were \(x-2\) and \(x-3\), we must exclude 2 and 3 from the domain of \(x\). Thus, along with the inequality above, we have the domain restrictions:

\(x \neq 2\) and \(x \neq 3\)

Combining these, the final solution is:

\(x < \frac{2}{3}\), \(x \neq 2\), \(x \neq 3\)