It's about rational Inequalities ​

To solve the rational inequality \frac{x-4}{x+2} > 0, we need to find the values of x for which the expression is positive. This is done by analyzing the signs of the numerator and the denominator.Step 1: Find the Critical PointsThe critical points are the values of x that make the numerator or the denominator equal to zero. * Numerator: x-4=0 \Rightarrow x=4 * Denominator: x+2=0 \Rightarrow x=-2These critical points divide the number line into three intervals: (-\infty, -2), (-2, 4), and (4, \infty).Step 2: Create a Sign TableWe'll test a value from each interval in the inequality to determine the sign of the expression in that interval.| Interval | Test Value | Sign of (x-4) | Sign of (x+2) | Sign of \frac{x-4}{x+2} ||---|---|---|---|---|| (-\infty, -2) | x=-3 | (-3-4) = - | (-3+2) = - | \frac{-}{-} = + || (-2, 4) | x=0 | (0-4) = - | (0+2) = + | \frac{-}{+} = - || (4, \infty) | x=5 | (5-4) = + | (5+2) = + | \frac{+}{+} = + |Step 3: Determine the SolutionThe inequality is \frac{x-4}{x+2} > 0, which means we are looking for intervals where the expression is positive. Based on the sign table, the expression is positive in the intervals (-\infty, -2) and (4, \infty).The solution to the inequality is the union of these two intervals.Solution: x \in (-\infty, -2) \cup (4, \infty)

Answered by armamentomelanie | 2025-08-23