Solving a Fractional Inequality Problem
<p>The provided inequality is: </p>
<p>\(\frac{4}{x-2} > \frac{7}{x-3}\)</p>
<p>To solve it, first find a common denominator: </p>
<p>\((x-2)(x-3)\)</p>
<p>Then multiply both sides by the common denominator, being careful to consider that it can change sign based on the value of \(x\): </p>
<p>\(4(x-3) > 7(x-2)\)</p>
<p>Now distribute and move all terms to one side: </p>
<p>\(4x - 12 > 7x - 14\)</p>
<p>Subtract \(4x\) and add \(14\) to both sides: </p>
<p>\(-12 + 14 > 7x - 4x\)</p>
<p>Simplify: </p>
<p>\(2 > 3x\)</p>
<p>Now, divide by 3: </p>
<p>\(\frac{2}{3} > x\)</p>
<p>So, the solution in terms of \(x\) is: </p>
<p>\(x < \frac{2}{3}\)</p>
<p>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: </p>
<p>\(x \neq 2\) and \(x \neq 3\)</p>
<p>Combining these, the final solution is: </p>
<p>\(x < \frac{2}{3}\), \(x \neq 2\), \(x \neq 3\)</p>
