Questions: (x+5)(x+1)/(x-4) ≥ 0 [?] ≤ x ≤ □ or x > □

Solution Steps

To solve the inequality \(\frac{(x+5)(x+1)}{x-4} \geq 0\), we need to determine the critical points where the expression is zero or undefined. These points will help us divide the number line into intervals to test the sign of the expression in each interval. The critical points are \(x = -5\), \(x = -1\), and \(x = 4\). We then test the sign of the expression in each interval created by these points.

To solve the inequality \(\frac{(x+5)(x+1)}{x-4} \geq 0\), we first find the critical points where the expression is zero or undefined. The critical points are determined by setting the numerator and denominator to zero:

• \(x + 5 = 0 \Rightarrow x = -5\)
• \(x + 1 = 0 \Rightarrow x = -1\)
• \(x - 4 = 0 \Rightarrow x = 4\)

Thus, the critical points are \(x = -5\), \(x = -1\), and \(x = 4\).

Next, we divide the number line into intervals based on the critical points:

1. \((- \infty, -5)\)
2. \((-5, -1)\)
3. \((-1, 4)\)
4. \((4, \infty)\)

We then select test points from each interval to determine the sign of the expression in those intervals:

• For \((- \infty, -5)\), we test \(x = -6\): the expression is negative.
• For \((-5, -1)\), we test \(x = -3\): the expression is positive.
• For \((-1, 4)\), we test \(x = 0\): the expression is negative.
• For \((4, \infty)\), we test \(x = 5\): the expression is positive.

From our tests, we find that the expression is non-negative in the intervals:

• \((-5, -1)\)
• \((4, \infty)\)

Final Answer