Question: Solve the rational inequality. Express your answer in interval notation. \[ -2x > \frac{4}{x -…

Solve the rational inequality. Express your answer in interval notation.

\[ -2x > \frac{4}{x - 3} \]

Answer

Solution

Step 1: Rewrite the inequality to have zero on one side. \[ -2x - \frac{4}{x - 3} > 0 & \] Step 2: Find a common denominator to combine the terms. \[ \frac{-2x(x - 3) - 4}{x - 3} > 0 & \] Step 3: Expand and simplify the numerator. \[ \frac{-2x^2 + 6x - 4}{x - 3} > 0 & \] Step 4: Factor the numerator if possible. \[ \frac{-2(x^2 - 3x + 2)}{x - 3} > 0 & \] Step 5: Factor the quadratic expression in the numerator. \[ \frac{-2(x - 1)(x - 2)}{x - 3} > 0 & \] Step 6: Determine the critical points from the numerator and denominator. Critical points are \( x = 1 \), \( x = 2 \), and \( x = 3 \). Step 7: Analyze the sign of the expression in each interval determined by the critical points. - For \( x < 1 \): All factors negative. The expression is negative. - For \( 1 < x < 2 \): \( (x - 1) \) positive, \( (x - 2) \) negative, \( (x - 3) \) negative. The expression is positive. - For \( 2 < x < 3 \): \( (x - 1) \) and \( (x - 2) \) positive, \( (x - 3) \) negative. The expression is negative. - For \( x > 3 \): All factors positive. The expression is positive. Step 8: Determine where the expression is greater than zero. The expression is positive in the intervals \( (1, 2) \) and \( (3, \infty) \). Step 9: Express the solution in interval notation. \( (1, 2) \cup (3, \infty) \)