Example Question - rational equation

Solving a Rational Equation

<p>Given equation:</p> <p>\(\frac{x}{2} = \frac{5x - 24}{x - 4}\)</p> <p>To solve for \( x \), cross-multiply to eliminate the denominators:</p> <p>\(x(x - 4) = 2(5x - 24)\)</p> <p>Expand both sides:</p> <p>\(x^2 - 4x = 10x - 48\)</p> <p>Move all terms to one side to set the equation to zero:</p> <p>\(x^2 - 4x - 10x + 48 = 0\)</p> <p>Simplify by combining like terms:</p> <p>\(x^2 - 14x + 48 = 0\)</p> <p>Factor the quadratic equation:</p> <p>\((x - 6)(x - 8) = 0\)</p> <p>Set each factor equal to zero:</p> <p>\(x - 6 = 0 \quad \text{or} \quad x - 8 = 0\)</p> <p>Solve for \( x \):</p> <p>\(x = 6 \quad \text{or} \quad x = 8\)</p>