Use the values determined in the previous steps to divide a number line into intervals.
Test a value in each interval to determine which intervals contain values that satisfy the inequality.
Let's deal with these steps one at a time.
Excluded Values
Consider the given inequality.
x-2/x+2+1/x-2>x-4/x-2Let's find the values for which any of the denominators is 0.
c|c
x+2=0 & x-2=0
⇕ & ⇕
x=-2 & x=2
The excluded values for this inequality are 2 and - 2.
Solve the Related Equation
We obtain the related equation by replacing the inequality sign with an equals sign.
c|c
Inequality & Equation [0.8em]
x-2/x+2+1/x-2 > x-4/x-2 & x-2/x+2+1/x-2 = x-4/x-2
Let's solve the equation. First, we will get rid of all of the denominators.
We will divide the number line considering the excluded values x=-2, x=2, and the solution to the related equation x=14.
Test Values
We will now test a value from each interval. Let's start with the interval x<- 2. We will test the point x=- 3. If substituting this point into the original inequality produces a true statement, then the numbers in this interval are solutions. If not, these numbers are not solutions.
Since we obtained a true statement, we can conclude that the numbers less than - 2 are solutions to the inequality. Using the same process, we can determine whether the numbers that belong to the other intervals are solutions.
Interval
Test Value
Statement
x< - 2
x=- 3
24>5 ✓
- 2< x < 2
x=0
-3>4 *
2
x=5
46>27 ✓
x>14
x=15
107>108 *
We found that the solutions to the given inequality are all the numbers below - 2 or all the numbers between 2 and 14.
Solution
x<-2 or 2
