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.
3/4-1/x-3 > x/x+4
Let's find the values for which any of the denominators is 0.
c|c
x-3=0 & x+4=0
⇕ & ⇕
x=3 & x=- 4
The excluded values for this inequality are 3 and - 4.
Solve the Related Equation
We obtain the related equation by replacing the inequality sign with an equals sign.
c|c
Inequality & Equation [0.8em]
3/4-1/x-3 > x/x+4 & 3/4-1/x-3 = x/x+4
Let's solve the equation. First, we will get rid of all of the denominators.
We obtained a quadratic equation. Let's identify the values of a, b, and c.
x^2-11x+52=0 ⇕ 1x^2+( - 11)x+ 52=0
We have that a= 1, b= - 11, and c= 52. We will solve this equation by substituting these values into the Quadratic Formula.
Since the radicand is a negative number, the equation does not have real solutions.
Divide the Number Line Into Intervals
Since the related equation does not have real solutions, we will divide the number line considering only the excluded values x=- 4 and x=3.
Test Values
We will now test a value from each interval. Let's start with the interval x<- 4. We will test the point x=- 5. 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 false statement, we can conclude that the numbers less than - 4 are not 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< - 4
x=- 5
0.875>5 *
- 4< x < 3
x=0
1.083>0 ✓
x>3
x=5
0.25>0.556 *
We found that the solutions to the given inequality are all the numbers between - 4 and 3.
Solution
- 4
