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.
1/4c+1/9c<1/2
Let's find the values for which any of the denominators is 0.
c|c
4c=0 & 9c=0
⇕ & ⇕
c=0 & c=0
The excluded value for this inequality is 0.
Solve the Related Equation
We obtain the related equation by replacing the inequality sign with an equals sign.
c|c
Inequality & Equation [0.8em]
1/4c+1/9c < 1/2 & 1/4c+1/9c = 1/2
Let's solve the equation. To do so, we will start by writing 4 as the product of 2 and 2 to help us finding the least common denominator (LCD).
1/4c+1/9c = 1/2
⇕
1/2( 2) c+1/9 c = 1/2
The least common denominator is the product of 2, 2, c, and 9.
Least Common Denominator
2( 2)( c)( 9) ⇔ 36c
To clear denominators, we will solve the equation by multiplying each side by the least common denominator.
We will divide the number line considering the excluded value c=0 and the solution to the related equation c= 1318.
Test Values
We will now test a value from each interval. Let's start with the interval c<0. We will test the point c=- 1. 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 0 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
c< 0
c=- 1
- 13<18 ✓
0< c < 13/18
c=1/36
13<0.5 *
c>13/18
c=1
13<18 ✓
We found that the solutions to the given inequality are all the numbers less than 0 or larger than 1318.
Solution
c<0 or 13/18
