Solve the rational inequality. Write your answer in interval notation. \( \frac{(x+12)(x-4)}{x-1} \geq 0 \)

To solve the inequality \( \frac{(x+12)(x-4)}{x-1} \geq 0 \), we first identify the critical points by setting the numerator and denominator to zero: 1. **Numerator**: \( (x+12)(x-4) = 0 \) gives \( x = -12 \) and \( x = 4 \). 2. **Denominator**: \( x - 1 = 0 \) gives \( x = 1 \). The critical points are \( x = -12, 1, \) and \( 4 \). Next, we use these points to test intervals on the number line: - Interval \( (-\infty, -12) \): Choose \( x = -13 \) → \(\frac{(-13+12)(-13-4)}{-13-1} = \frac{(-1)(-17)}{-14} > 0\). - Interval \( (-12, 1) \): Choose \( x = 0 \) → \(\frac{(0+12)(0-4)}{0-1} = \frac{(12)(-4)}{-1} > 0\). - Interval \( (1, 4) \): Choose \( x = 2 \) → \(\frac{(2+12)(2-4)}{2-1} = \frac{(14)(-2)}{1} < 0\). - Interval \( (4, \infty) \): Choose \( x = 5 \) → \(\frac{(5+12)(5-4)}{5-1} = \frac{(17)(1)}{4} > 0\). Now we summarize the signs: - \( (-\infty, -12) \): positive - \( (-12, 1) \): positive - \( (1, 4) \): negative - \( (4, \infty) \): positive To satisfy \( \frac{(x+12)(x-4)}{x-1} \geq 0 \), we include the intervals where the expression is positive, as well as the points where the expression equals zero. Thus: - At \( x = -12 \) and \( x = 4 \), the expression is zero. - At \( x = 1\), the expression is undefined. Combining this, we find the intervals satisfying the inequality: The solution in interval notation is: \[ (-\infty, -12] \cup (-12, 1) \cup (4, \infty) \]