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/5+2/3<3/x-4
Let's find the values for which any of the denominators is 0.
x-4=0
⇕
x=4The excluded value for this inequality is 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]
x/5+2/3 < 3/x-4 & x/5+2/3 = 3/x-4
Let's solve the equation. First, we will get rid of all of the denominators by multiplying the equation by the least common denominator (LCD). The LCD of the given equation in 3(5)(x-4).
We obtained a quadratic equation. Let's identify the values of a, b, and c.
3x^2-2x-85=0 ⇕ 3x^2+( - 2)x+( -85)=0
We have that a= 3, b= - 2, and c= -85. We will solve this equation by substituting these values into the Quadratic Formula.
We will find the solutions by dividing our answer in two using the positive and negative sign.
x=2± 32/6
x=2- 32/6
x=2+ 32/6
x=-30/6
x=34/6
x=-5
x=17/3
The solutions to the equation are x=-5 and x= 173.
Divide the Number Line Into Intervals
We will divide the number line considering the excluded value x= 4 and the solutions to the related equation x=-5 and x= 173.
Test Values
We will now test a value from each interval. Let's start with the interval x<- 5. We will test the point x=- 10. 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 - 5 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< - 5
x=- 10
56 > 9 ✓
- 5< x < 4
x=0
8<- 9 *
4
x=5
4 <9 ✓
x>17/3
x=10
16 <3 *
We found that the solutions to the given inequality are all the numbers less than - 5, or between 4 and 173.
Solution
x<-5 or 4
