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 points by setting the numerator and denominator equal to zero.
(x - 4) = 0 and (-x + 2) = 0
Solving these equations, we find:
x = 4 and x = 2
Step 2: Create a number line and mark the critical points on it.
--------------2---------4-------------
Step 3: Test intervals between the critical points.
Choose a test value in each interval and substitute it into the inequality. If the result is positive, the interval is part of the solution.
For the interval (-∞, 2), we can choose x = 1:
((1 - 4) / (-(1) + 2)) > 0
(-3 / 1) > 0
-3 > 0
The result is negative, so the interval (-∞, 2) is not part of the solution.
For the interval (2, 4), we can choose x = 3:
((3 - 4) / (-(3) + 2)) > 0
(-1 / -1) > 0
1 > 0
The result is positive, so the interval (2, 4) is part of the solution.
For the interval (4, ∞), we can choose x = 5:
((5 - 4) / (-(5) + 2)) > 0
(1 / -3) > 0
-1/3 < 0
The result is negative, so the interval (4, ∞) is not part of the solution.
Step 4: Determine the solution based on the test results.
The solution to the inequality is the interval (2, 4), excluding the critical points.
Therefore, using interval notation, we can write the solution as:
(2, 4)