Precalculus : Solve and Graph Rational Inequalities

Study concepts, example questions & explanations for Precalculus

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Solving Polynomial And Rational Inequalities

Solve the inequality. 

Possible Answers:

Correct answer:

Explanation:

First, subtract  from both sides so you get 

.

Then find the common denominator and simplify 

.

Next, factor out the numerator 

 

and set each of the three factor equal to zero and solve for .

The solutions are 

.

Now plug in values between , and  into the inequality and observe if the conditions of the inequality are met.

Note that . They are met in the interval  and .

Thus, the solution to the inequality is  

Example Question #3 : Solving Polynomial And Rational Inequalities

Solve and graph:

Possible Answers:

Correct answer:

Explanation:

1) Multiply both sides of the equation by the common denominator of the fractions:

 

     

     

2) Simplify:

     

     

     

     

     

3) For standard notation, and the fact that inequalities can be read backwards:

    

 

     For interval notation:

      

4) Graph:

 Varsitytutors5

Example Question #4 : Solving Polynomial And Rational Inequalities

Solve and graph:

Possible Answers:

Correct answer:

Explanation:

Graph the rational expression,

1) Because  and a divide by is undefined in the real number system, there is a vertical asymptote where .

2) As   ,   , and as  ,  .

3) As  ,  , and as   ,  .

4) The funtion y is exists over the allowed x-intervals:

 Varsitytutors6

 

One approach for solving the inequality: 

For 

1) Determine where  over the x-values  or .

2)  for the intervals  or .

3) Then the solution is .

 

Another approach for solving the inequality:

1) Write   as , then determine the x-values that cause  to be true: 

Varisty3

2)  is true for  or .

3) Then the solution is .

Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: