TABLE OF CONTENTS

1.3 Absolute Value Equations, Inequalities, and Functions

Prerequisite knowledge and skills:

        a working knowledge and understanding of absolute value of a number

Terms to know: 

·         Absolute value

·         Distance on number line

Skill Prep 1:  Distances on the number line

Skill Prep 2:  Factoring out the leading coefficient

Concept Prep:  Absolute value and graphs

Introduction

Try It!

Suppose the “average” score on a test is 100 points and 68% of the population is within 15 points of that score either way.  What is the range of scores of 68% of the population?

Write a mathematical expression using symbols such as   to represent this situation if s is the score on the test.

Write a mathematical expression using absolute value notation  to represent this scenario if s is the score on the test.

Try It!

Suppose that a number line represents Main Street in Kent with units measured in “city blocks.”  Suppose also that the origin represents KSU.   If you are at  or , how many blocks are you away from KSU? 

Write a mathematical expression using absolute value notation  to represent this scenario.

Clearly you are 3 blocks away from KSU; in both cases your distance from the origin is 3.   We use absolute value  to represent distance from the origin.   Here   Whether you are three blocks east or west, you still need to walk 3 blocks to get to KSU.  Distance is always positive or zero, never negative. 

If you were 4 blocks from KSU, your position x satisfies .    What are the possible values of x ? 

Where might you be if your position, x, is described as  blocks?  Sketch your solution on the number line below. 

                     <                                                >        

What if  ?    Sketch your solution on the number line below. 

                     <                                                >        

Your graphs should look like this.

Notice that for , the numbers fall in the interval .

For  the numbers fall in either of the intervals .

Experiment with the examples in the following checkpoint exercise, then try to generalize your answer.  What patterns did you find?

    Checkpoint 1

We can think of the absolute value  of a number x as its distance from the origin on the number line.  We write , but can also think of it as .

Suppose you were a distance of four blocks from a point two blocks east of the origin.  Where could you be?  Mark your possible positions on the number line below.

                     <                                                >        

Mathematicians would write   to represent this situation.

Suppose you were less than four blocks from a point two blocks east of the origin.  Where could you be?  Mark your possible positions on the number line below.

                     <                                                >        

Mathematicians would write   to represent this situation.

Finally, if you were more than four blocks from a point two blocks east of the origin, where would you be?  Mark your possible positions on the number line below.

                     <                                                >        

Mathematicians would write   to represent this situation.

Just as  or  is the distance of x from the origin on the number line,  is the distance from the number 2 on the number line.   In general,

 is the distance from a on the number line.

     Checkpoint 2

We can use the above summary to help us solve equations and inequalities involving absolute values.

Example 1.  Solve     

Write an equivalent statement:      or   

Solve for x:                  or       

                                                      or       

Check:                         or      

Check by graphing

Example 2.  Solve     

Write an equivalent statement:      

Think:  we want numbers whose distance from 5 is LESS THAN 8.

Solve for x:                   

Write as an interval:                    

Check on the number line:  Which numbers are less than 8 units away from 5 on the number line?

Check by graphing

Example 3   

Write an equivalent statement:       

Think:  we want numbers whose distance from 5 is GREATER THAN 8. 

Solve for x:                   

Write using intervals:                    

Check on the number line:  Which numbers are more than 8 units away from 5 on the number line?

Check by graphing

Example 4   

Method 1

Write an equivalent statement:        

Solve for x:                                          

Write using intervals:                    

Check by graphing

Method 2

If we want to think about our number line model, we could factor out the leading coefficient, like so:

Divide through by this leading coefficient:     .

Now we can use our number line model:

Think:  we want numbers whose distance from  is GREATER THAN 4.  You might want to sketch a number line to help you think it through.

Solve for x:                                     

Write using intervals:                            

Which is the same answer we obtained previously.

Example 5   

If we’re really on the ball here and paying close attention, we can save ourselves some work!!  (That’s always a good thing.)

Notice that this inequality is looking for values that make an expression in absolute value LESS THAN 0.  This can never be!  So we’re done!  There is NO SOLUTION to this inequality. 

More Worked Examples: Equations

More Worked Examples: Inequalities  

Homework

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