Questions: Solve the following inequality. [ frac-121-x>x-2 ] Write your answer as an interval or union of intervals. If there is no real solution, click on "No solution".

Solution Steps

To solve the inequality \(\frac{-12}{1-x} > x - 2\), we need to follow these steps:

1. Identify the critical points by setting the denominator and the expression on the right-hand side to zero.
2. Determine the intervals created by these critical points.
3. Test each interval to see where the inequality holds true.
4. Combine the intervals where the inequality is satisfied.

We start with the inequality: \[ \frac{-12}{1 - x} > x - 2 \]

To solve the inequality, we first find the critical points by setting the denominator and the right-hand side to zero. The critical points occur when:

1. \(1 - x = 0 \Rightarrow x = 1\)
2. \(x - 2 = 0 \Rightarrow x = 2\)

The critical points divide the number line into the following intervals:

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

We will test each interval to see where the inequality holds true.

1. For \(x < -2\): Choose \(x = -3\): \[ \frac{-12}{1 - (-3)} = \frac{-12}{4} = -3 \quad \text{and} \quad -3 > -5 \quad \text{(True)} \]

2. For \(1 < x < 5\): Choose \(x = 3\): \[ \frac{-12}{1 - 3} = \frac{-12}{-2} = 6 \quad \text{and} \quad 6 > 1 \quad \text{(True)} \]

3. For \(x > 5\): Choose \(x = 6\): \[ \frac{-12}{1 - 6} = \frac{-12}{-5} = \frac{12}{5} \quad \text{and} \quad \frac{12}{5} > 4 \quad \text{(False)} \]

The inequality holds true in the intervals:

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

Final Answer