Solve the following inequality.
(x-4)/(-x+2)>0
Write your answer using interval notation
To solve the inequality (x-4)/(-x+2) > 0, we can follow these steps:
Step 1: Find the critical values.
The inequality is based on a rational expression, so we need to identify the values of x that make the numerator or denominator equal to zero. In this case:
x - 4 = 0 => x = 4 (numerator equals zero)
-x + 2 = 0 => x = 2 (denominator equals zero)
Step 2: Create a number line.
Now, draw a number line and place these critical values on it. It will look like this:
-∞ |------(2)------(4)------|
Step 3: Choose test intervals.
We need to test intervals on either side of the critical values to determine the solution. Let's choose three intervals:
- For x < 2, choose a value less than 2, like x = 0.
- For 2 < x < 4, choose a value between 2 and 4, like x = 3.
- For x > 4, choose a value greater than 4, like x = 5.
Step 4: Evaluate the expression in each interval.
Now, substitute the chosen test values into the inequality expression, (x-4)/(-x+2), and check the sign of the result to determine if it is greater than zero (+) or less than zero (-).
- For x = 0:
[(0-4)/(-0+2)] = (-4)/2 = -2, which is less than zero (-).
- For x = 3:
[(3-4)/(-3+2)] = (-1)/(-1) = 1, which is greater than zero (+).
- For x = 5:
[(5-4)/(-5+2)] = (1)/(-3) = -1/3, which is less than zero (-).
Step 5: Determine the solution.
Based on the signs obtained in Step 4, we can determine that the solution is where the inequality expression is greater than zero (+). Therefore, the solution interval is (2, 4).
Writing the solution in interval notation, we represent the solution as:
(2, 4)