Solve the rational inequality \(\frac{x^2 - 4}{x^2 - 5x + 6} \geq 0 \)
Answer
Finally, the solution to the inequality \(\frac{x^2 - 4}{x^2 - 5x + 6} \geq 0\) is the union of the intervals where the value of the rational expression is positive, as well as the points where the expression is zero.
Step 1 ：First, factorize the numerator and denominator: \(\frac{(x-2)(x+2)}{(x-2)(x-3)} \geq 0 \)
Step 2 ：Next, find the critical points by setting the numerator and denominator equal to zero: Critical points are \(x = -2, 2, 3\)
Step 3 ：Then, test the intervals \((-\infty, -2)\), \((-2, 2)\), \((2, 3)\), and \((3, \infty)\) by choosing a test point from each interval and evaluating the sign of \(\frac{(x-2)(x+2)}{(x-2)(x-3)}\)
Step 4 ：For \(x = -3\), \(x = 0\), \(x = 2.5\), and \(x = 4\), we get +, -, +, and - respectively.
Step 5 ：Finally, the solution to the inequality \(\frac{x^2 - 4}{x^2 - 5x + 6} \geq 0\) is the union of the intervals where the value of the rational expression is positive, as well as the points where the expression is zero.
