TABLE OF CONTENTS

2.4 Solving Quadratic Inequalities

 

Prerequisite knowledge and skills

*       A working knowledge and understanding of solving quadratic equations

*       A working knowledge and understanding of graphs of quadratic functions:  x-intercepts, intervals of positive and negative function values

 

Terms to know:  

·         factor

·         inequality

·         interval

·         leading coefficient

·         x-intercepts

·         zero of a function

 

Introduction

Skill prep:  Determining which values make an expression positive, negative, or zero

Concept prep assignment:  Determining intervals for which a graph is positive, negative or 0.

Skill prep:  Interval notation

 

 

 

2.4.1 Solutions from graphs

Let’s look at some of the graphs from the concept prep assignment.

 

 

1.

 

The function, f, given by  is

a possible formula for this function.  Notice

that the function outputs are negative when the

input values are in the interval .

 

The output values are positive when the output

values are in the intervals    or   

 

Negative function outputs here.  (Function outputs are the y-coordinates)             

 

 

 

2.                                                             

 

This parabola has no x-intercepts and only positive output, or y values.  We say the function values are positive  in the interval  and never negative.

 

 

 

 

 

 

 

 

3.                                                                   

The x-intercepts are -1 and 3, so the factors are   and .   The function can be given by .  (Note that this formula results in the correct vertical intercept.  This might  not always happen.  We  sometimes might need a leading coefficient other than one.)                       

                                                                                                                       

 

The function outputs are positive  when the

 input values are in the interval .                                                         

                                                                     

 

 

The function outputs are negative when the input values are in the intervals    or  .

                                                                

 

 

Checkpoint 1

 

  

2.4.2 Algebraic Solutions

 

            In practice it is helpful to know how to solve these inequalities algebraically in addition to graphically.  You may not always have a graphing utility handy; knowing more than one way of solving it deepens your understanding. 

 

 

Example 1.  Solve  algebraically. 

 

1.  Get one side equal to 0.        Already done:        

 

2.  Factor the quadratic.           Already done:         

 

3.  Find the zeros of the function:        =  0, so  

                                                                 =  0, so  x = 2.

 

 

These numbers are sometimes called critical numbers because the function may change from positive to negative (or vice versa) at these values of the input. 

 

Now think a minute.   We need to find out for which inputs (x -values) are the outputs (the y values) negative.  There are several ways of doing this.  Before reading on, try to think of a way to solve this problem.

 

 

 

 

 

 

 

 

 

 

 

 

 

Recall the graph of this quadratic function:

 

If the function outputs are equal to 0 at

 and 2, what do you notice about the values

of the output in between   and 2 ?

 

Do you see how they all have the same sign, 

in this case negative?

 

Do you see that all the outputs to the left of

 also have the same sign  all positive?

 

Do you see that all the outputs to the right of

2 also have the same sign  also all positive?

 

 

One way of organizing this information algebraically is to use something called a sign chart.

A sign chart uses the number line with the critical numbers (the zeros of the function) as endpoints of intervals.  The beginning of a sign chart for the problem above would look like this:

 

4.      Create a sign chart.

 

 

 

                                

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

       

 

 

 

 

 

 

We plan to determine the sign (positive or negative) of each factor in each of these intervals, so we write each factor on the left side of the number line and the complete factored form near the bottom,  like this: 

 

5.      Write each factor on the sign chart.

 

             

 

          

 

 

 

 

 

 

 

 

 

We will analyze each factor separately using test numbers.

 

 

 

 

6.         Choose ANY NUMBER in an interval as a test number. 

 

Recall that the function changes sign only at the critical numbers.   Recall also that our intervals  have these critical numbers as endpoint.  This means that all inputs in each  interval will have outputs with the same sign.

           

(THINK ABOUT THIS LAST SENTENCE UNTIL IT MAKES SENSE TO YOU. REREAD IT OVER AND OVER UNTIL YOU UNDERSTAND WHAT IT’S SAYING.  REMEMBER WHAT THE GRAPHS LOOK LIKE.  TALK TO YOUR GROUP MEMBERS ABOUT IT.)

 

 

       

 

                  Left interval:  :          If  

                                                                                 

                  Middle interval: :             If x = 0, then  , which is POSITIVE                                                      

                                                                                and   =  -2, which is NEGATIVE.

              

                 Right interval:    :            If x = 3, then , which is POSITIVE

                                                                    and    = 3, which is POSITIVE.

 

7.      Summarize your work on the sign chart by placing the appropriate sign (  ) by each factor.   Multiply the signs together to obtain the sign of the product.

 

         (x+4)             -                +              +                                 

        (x-2)              -                -              +

 

(x+4)(x-2)             +            -          +           

 

 

 

 

   Can you see which interval is the solution?  For which x values are the outputs less than 0?

 

 

 

 

 

The solution interval is (-4,2).  

 

 

This solution is consistent with our graphical solution.  Of course, checking by graphing is always a good idea. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Steps for solving a quadratic inequality algebraically:  

 

          1.  Move all terms to one side of the polynomial so that the other side = 0.

          2.  Factor the polynomial and find the critical numbers.

          3.  Make a sign chart with these critical numbers as interval endpoints.

               List the factors to the left.

          4.  Analyze the sign of each factor in each interval using test numbers.

          5.  Multiply the signs together in each interval and decide upon the solution

                interval(s).

 

 

 

 

 

 

 

 

 

Example 2.  Solve    

 

1.       Get one side = 0:                                     so   

2.      Factor and find the critical numbers.    so the critical numbers   

                                                                   are x = 5 and x = -3.

 

3.      Put the critical numbers on a sign chart and list the factors to the left.

         

         

 

 

 

 

 

 

 

 

4.      Analyze the sign of each factor in each interval using test numbers.

                   -                 -                +  

                   -                          +                       +

 

 

 

 

 

 

   If x = -4, then  is NEGATIVE

                   and  is NEGATIVE .

 

  If x = 0 ,  then  is still NEGATIVE

                   and  is POSITIVE

 

 If x = 6 ,  then  is POSITIVE

                  and  is POSITIVE.

 

5.  Multiply the signs together in each interval and

 

6.  Decide upon the solution interval.

                   -                 -                +  

                   -                          +                       +

 

        +              -              +

 

 

 

The solution intervals are   or   because the inequality we are solving desires outputs that are , or positive.    Sometimes you will see the  symbol instead of the word “or:”  .

Be sure to check that graphical representation of the function supports our answer.

 

 

 

Example 3.  Solve the inequality .  This inequality is not a quadratic, but the procedure is the same.

 

1.       Get one side = 0:                            so     

 

 

2.      Factor and find the critical numbers.    

                                                                                                                       

      The critical numbers  are .

 

3.      Put the critical numbers on a sign chart and list the factors to the left.

 

                                                      

                                         

          

 

       

 

 

 

4.      Analyze the sign of each factor in each interval using test numbers.

                   -        +     +      +                                         

                     -             -        +         +                             

                 -             -       -          +

 

       

 

 

 

          These details are left for you.  Which test numbers did you pick?  Do your signs agree with those in the above chart?

 

 

5.  Multiply the signs together in each interval and

 

6.  Decide upon the solution interval.

                  -        +     +      +                                         

                     -             -       +         +                             

                 -             -       -          +

      -     +           -       +  

                                        

                                                             

 

Less than zero means look for intervals with negative product.

 

 

 

The solution to  is  , since these are the intervals with NEGATIVE results. 

 

Check to see if the graphical representation supports our solution.

 

More detail

 

More worked examples

 

 

Checkpoint 2

 

Homework problems

 

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