Solve the following inequality.
(x-4)/(-x+2)>0
Write your answer using interval notation
To solve the inequality (x-4)/(-x+2) > 0, you can use the following steps:
Step 1: Find the critical values:
Set the denominator, -x+2, equal to 0 and solve for x:
-x+2 = 0
x = 2
Step 2: Determine the sign of the expression (x-4)/(-x+2) for each interval:
Consider three intervals:
Interval 1: x < 2
Interval 2: x > 2
Interval 3: x = 2
For Interval 1 (x < 2):
Choose a value less than 2, for example x = 1.
Evaluate the expression: (1-4)/(-(1)+2) = -3/1 = -3.
Since -3 is negative, the expression is negative for Interval 1.
For Interval 2 (x > 2):
Choose a value greater than 2, for example x = 3.
Evaluate the expression: (3-4)/(-(3)+2) = -1/-1 = 1.
Since 1 is positive, the expression is positive for Interval 2.
For Interval 3 (x = 2):
Substitute x = 2 into the expression: (2-4)/(-(2)+2) = -2/0.
Note that division by zero is undefined, so Interval 3 is excluded from the solution.
Step 3: Write the solution using interval notation:
Since the expression (x-4)/(-x+2) is greater than 0, it is positive for x < 2 and x > 2. However, note that x = 2 is excluded from the solution. Hence, the solution in interval notation is (-∞, 2) ∪ (2, +∞).