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College Algebra
Tutorial 23B: Rational Inequalities


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Solve rational inequalities using a sign graph of factors.
  2. Solve rational inequalities using the test-point method.




deskIntroduction



In this tutorial we will be looking at solving rational inequalities using two different methods.  The methods of solving rational inequalities are very similar to solving quadratic inequalities.  If you need a review on solving quadratic inequalities, feel free to go to Tutorial 23A: Quadratic Inequalities.  And yes, we will be dealing with fractions (yuck!) as we go through the rational inequalities.  I think we are ready to start.

 

 

desk Tutorial



 

Rational Inequalities

A rational inequality is one that can be written in one of the following standard forms:

rational expression
or
rational expression
or
rational expression
or
rational expression

Q does not equal 0.



In other words, a rational inequality is in standard form when the inequality is set to 0.

 
  
Solving Rational Inequalities
Using a Sign Graph of the Factors

 
This method of solving rational inequalities only works if the numerator and denominator factor.  If at least one of them doesn't factor then you will need to use the test-point method shown later on this page.

This method works in the same fashion as it does with quadratic inequalities.

If you need a review on solving quadratic inequalities, feel free to go to Tutorial 23A: Quadratic Inequalities.

Be careful, it is really tempting to multiply both sides of the inequality by the denominator like you do when solving rational equations.  The problem is the expression in the denominator will have a variable, so we won't know what the denominator is equal to.  Remember that if we multiply both sides of an inequality by a positive number, it does not change the inequality.  BUT if we multiply both sides by a negative, it does change the sign of the inequality.  Since we don't know what sign we are dealing with we need to go about it the way described below.


 

Step 1: Write the rational inequality in standard form.

 
It is VERY important that one side of the inequality is 0.

0 is our magic number.  It is the only number that separates the negatives from the positives.  If an expression is greater than 0, then there is no doubt that its sign is positive.  Likewise, if it is less than 0, its sign is negative.  You can not say this about any other number.  Since we are working with inequalities, this idea will come in handy.  With this technique we will be looking at the sign of a number to determine if it is a solution or not.


 

Step 2: Factor the numerator and denominator and find the values of x that make these factors equal to 0 to find the boundary points.

 
The boundary point(s) will mark off where the rational expression is equal to 0.  This is like the cross over point.  0 is neither positive or negative.

As mentioned above, this method of solving rational inequalities only works if the numerator and denominator factor.  If at least one of them doesn't factor then you will need to use the test-point method shown later on this page.


 

Step 3: Use the boundary point(s) found in step 2 to mark off test intervals on the number line.

 
The boundary point(s) on the number will create test intervals.

 

Step 4: Find the sign of every factor in every interval. 

 

You can choose ANY value in an interval to plug into each factor.  Whatever the sign of the factor is with that value gives you the sign you need for that factor in that interval.  Make sure that you find the sign of every factor in every interval.

Since the inequality will be set to 0,  we are not interested in the actual value that we get when we plug in our test points, but what SIGN (positive or negative) that we get. 


 

Step 5: Using the signs found in Step 4, determine the sign of the overall rational function in each interval.


Since the inequality will be set to 0,  we are not interested in the actual value that we get when we plug in our test points, but what SIGN (positive or negative) that we get.

When you look at the signs of your factors in each interval, keep in mind that they represent a product and/or quotient of the factors that make up your overall rational function. 

You determine the sign of the overall rational function by using basic multiplication sign rules:

  • The product or quotient of two factors that have the same sign is positive.

  • The product or quotient of two factors that have the opposite signs is negative.

This can be extended if you have more than two factors involved.


If the rational expression is less than or less than or equal to 0, then we are interested in values that cause the rational expression to be negative.

If the rational expression is greater than or greater than or equal to 0, then we are interested in values that cause our rational expression to be positive.


 

Step 6: Write the solution set and graph.

 
If you need a review on writing interval notation or graphing an inequality, feel free to go to Tutorial 22: Linear Inequalities.


 
 

notebook Example 1: Solve, write your answer in interval notation and graph the solution set: example 1.

videoView a video of this example



 
Step 1: Write the rational inequality in standard form.

 
This rational inequality is already in standard form.

 
 
Step 2: Factor the numerator and denominator and find the values of x that make these factors equal to 0 to find the boundary points.

 
Numerator:

 
example 4a
*Set numerator = 0 and solve

 
Denominator:

 
example 4b
*Set denominator = 0 and solve

 
-5 and 1 are boundary points.

 

 
Below is a graph that marks off the boundary points -5 and 1 and shows the three sections that those points have created on the graph.  Note that there is a open hole at -5.  Since that is the value that causes the denominator to be 0, we cannot include where x = -5.  Since our inequality includes where it is equal to 0, and 1 causes only the numerator to be 0 there is a closed hole at 1.
 
 

blank

Note that the two boundary points create three sections on the graph: example 4eexample 4f, and example 4g.


 
 

 
You can choose ANY point in an interval to represent that interval.  Remember that we are not interested in the actual value that we get, but what SIGN (positive or negative) that we get.


If we chose a number in the first interval, example 4e, like -6 (I could have used -10, -25, or -10000 as long as it is in the interval), it would make both factors negative:

  -6 - 1 = -7 and -6 + 5 = -1


If we chose a number in the second interval, example 4f, like 0 (I could have used -4, -1, or 1/2 as long as it is in the interval), it would make x - 1 negative and x + 5 positive: 

0 - 1 = -1 and 0 + 5 = 5


If we chose a number in the third interval, example 4g, like 2 (I could have used 10, 25, or 10000 as long as it is in the interval), it would make both factors positive:

  2 - 1 = 1 and 2 + 5 = 7


example 1b1



In the first interval, example 4c, we have a negative divided by a negative, so the sign of the quadratic in that interval is positive.

In the second interval, example 4e, we have a positive divided by a negative, so the sign of the quadratic in that interval is negative.

In the third interval,  example 4f, we have two positives, so the sign of the quadratic in that interval is positive.

Keep in mind that our inequality is example 1b3.  Since we are looking for the quadratic expression to be GREATER THAN OR EQUAL TO 0, that means we need our sign to be POSITIVE (OR O).

example 1b2




 
Interval notation: example 4g

Graph: 
example 4k

*An open interval indicating all values less than -5  and a closed interval indicating all values greater then or equal to 1

*Visual showing all numbers less than -5 or greater then or equal to 1
 


 
 

Solving Rational Inequalities
Using the Test-Point Method

 
The test-point method for solving rational inequalities works for any rational function that has a real number solution, whether the numerator or denominator factors or not.

This method works in the same fashion as it does with quadratic inequalities.

If you need a review on solving quadratic inequalities, feel free to go to Tutorial 23A: Quadratic Inequalities.

Be careful, it is really tempting to multiply both sides of the inequality by the denominator like you do when solving rational equations.  The problem is the expression in the denominator will have a variable, so we won't know what the denominator is equal to.  Remember that if we multiply both sides of an inequality by a positive number, it does not change the inequality.  BUT if we multiply both sides by a negative, it does change the sign of the inequality.  Since we don't know what sign we are dealing with we need to go about it the way described below.


 

Step 1: Write the rational inequality in standard form.

 
It is VERY important that one side of the inequality is 0.

0 is our magic number.  It is the only number that separates the negatives from the positives.  If an expression is greater than 0, then there is no doubt that its sign is positive.  Likewise, if it is less than 0, its sign is negative.  You can not say this about any other number.  Since we are working with inequalities, this idea will come in handy.  With this technique we will be looking at the sign of a number to determine if it is a solution or not.


 

Step 2:  Find the values of x that make the numerator and denominator equal to 0 to find the boundary points.

 
The boundary point(s) will mark off where the rational expression is equal to 0.  This is like the cross over point.  0 is neither positive or negative. 

 

Step 3: Use the boundary point(s) found in Step 2 to mark off test intervals on the number line.

 
The boundary point(s) on the number will create test intervals.

 

Step 4: Test a point in each test interval found in Step 3 to see which interval(s) is part of the solution set.

 
You can choose ANY point in an interval to represent it.  You need to make sure that you test one point from each interval.  Sometimes more than one interval can be part of the solution set.

Since the inequality will be set to 0,  we are not interested in the actual value that we get when we plug in our test points, but what SIGN (positive or negative) that we get. 

If the rational expression is less than or less than or equal to 0, then we are interested in values that cause the rational expression to be negative.

If the rational expression is greater than or greater than or equal to 0, then we are interested in values that cause our rational expression to be positive.


 

Step 5: Write the solution set and graph.

 
If you need a review on writing interval notation or graphing an inequality, feel free to go to Tutorial 22: Linear Inequalities.
 

 
 

notebook Example 2: Solve, write your answer in interval notation and graph the solution set: example 2.

videoView a video of this example



 
Step 1: Write the rational inequality in standard form.

 
example 5b

*Inv. of add. 1 is sub. 1
 
 
 
 

 


 
 
Step 2: Factor the numerator and denominator and find the values of x that make these factors equal to 0 to find the boundary points.

 
Numerator:

 
example 5c
*Set numerator = 0 and solve
 
 
 
 
 

 


 
Denominator:

 
example 5d
*Set denominator = 0 and solve

 
-1/4 and 0 are boundary points.

 

 
Below is a graph that marks off the boundary points -1/4 and 0 and shows the three sections that those points have created on the graph.  Note that open holes were used on those two points since our original inequality did not include where it is equal to 0 and -1/4 makes the denominator 0. 
 
 

example 5e

Note that the two boundary points create three sections on the graph: example 5fexample 5g , and example 5h.


 
 
Step 4: Test a point in each test interval found in step 3 to see which interval(s) is part of the solution set.

 
You can choose ANY point in an interval to represent that interval.  Remember that we are not interested in the actual value that we get, but what SIGN (positive or negative) that we get.

Keep in mind that our original problem is example 5m.  Since we are looking for the quadratic expression to be LESS THAN 0, that means we need our sign to be NEGATIVE.

From the interval example 5f, I choose to use -1 to test this interval:
(I could have used -10, -25, or -10000 as long as it is in the interval)


 
example 5i
*Chose -1 from 1st interval to plug in for
 

 


 
Since 3 is positive and we are looking for values that cause our quadratic expression to be less than 0 (negative), example 5fwould not be part of the solution.

 
 
From the interval example 5g, I choose to use -1/5 to test this interval.
(I could have used -1/6, -1/7, or -1/8 as long as it is in the interval)

 
example 5j

*Chose -1/5 from 2nd interval to plug in for
 
 
 
 
 
 
 
 
 
 
 

 


 
Since -1 is negative and we are looking for values that cause our expression to be less than 0 (negative), example 5gwould be part of the solution.

 
 
From the interval example 5h, I choose to use 1 to test this interval.
(I could have used 10, 25, or 10000 as long as it is in the interval)

 
example 5k
*Chose 1 from 3rd interval to plug in for
 

 


 
Since 5 is positive and we are looking for values that cause our quadratic expression to be less than 0 (negative), example 5h would not be part of the solution.

 
 

 
Interval notation: example 5g

Graph: 
example 5l

*Open interval indicating all values between -1/4 and 0

*Visual showing all numbers between -1/4 and 0 on the number line

 

 

 

desk Practice Problems



These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1b: Solve (using any method), write your answer in interval notation and graph the solution set.


 

 

 

desk Need Extra Help on these Topics?



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Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

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Last revised on Jan. 2, 2010 by Kim Seward.
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