Solving Open Sentences Involving Absolute Value 7-6

7-6: Solving Open Sentences Involving Absolute Value 

OBJECTIVE: 

You will be able to solve open sentences involving absolute value and 

graph the solutions. 

We need to start with a discussion of what absolute value means. 

Absolute value is a means of determining the distance from zero. 

If I ask for the absolute value of 4, the answer is “4” since 4 is four units 

away from zero. 

If I ask for the absolute value of -4, the answer is “4” since -4 is four 

units away from zero. 

|4| = 4 |-4| = 4 

© William James Calhoun, 2001 


Now examine absolute values with variables in them. 

|x| = 4 can be rewritten as: 

|x - 0| = 4 

Rewriting the absolute value this way helps to explain what is going on. 

The absolute value tells us, “The distance between x and zero is four 

units.” 

So, x if four units away from zero in either direction. 

Graphically: 

4 units 4 units 

In non-graphical terms, |x - 0| = 4 tells us: 

x = 4 

or 

-6 -4 -2 0 2 4 6 

x = -4 


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EXAMPLE 1: Solve |x - 3| = 5. 

Use the definition of absolute value. 

This problem is saying, “The distance between x and 3 is 5 units.” 

Graphically: 

Run both directions 5 units. 


-3 -1 1 3 5 7 9 

Center on the 3. 

The solution is then: 

x = -2 or x = 8 

{-2, 8} 

Another way to solve this problem without graphing it first follows. 

The problem tells us that the distance between x and 3 is 5 units, so we know either: 

x - 3 = 5 

+3 +3 

or 

solve 

x - 3 = -5 

+3 +3 

x = 8 

these 

x = -2 

equations 

The solution is then: 

x = -2 or x = 8 

{-2, 8} 

This second method is the CPM and is how I recommend you work these problems. 



Now, what if we introduce inequalities into the absolute value mix. 

What does |x| < n mean? 

Again, this can be rewritten as: 

|x - 0| < n 

This tells us that, “The distance between x and zero is less than n.” 

Using a real number for n, we can see graphically what is going on. 

|x| < 4 


Run four units to the right, but 

aka 

do not include the number four 

|x - 0| < 4 

since it is not “or equal to.” equal to.” 


Since the problem says the 

distance is “less than 4,” we 

need to shade everything 

between the center line and 4. 

Now, run four units to the left, 

but do not include the number 

negative four since it is not “or 

Since the problem says the distance 

is “less than 4,” we need to shade 

everything between the center line 

and -4. 

-6 -4 -2 0 2 4 6 

The answer to the inequality, |x| < 4 is then: 

{x | -4 < x < 4} 

If |x| < 4 then we can see that either: 

x < 4 AND x > -4 

We will use this to solve inequalities. 


2


EXAMPLE 2: Solve |3 + 2x| < 11 and graph the solution set. 

Use what we just saw to rewrite the problem as two inequalities connected with “and.” 

3 + 2x < 11 and 3 + 2x > -11 

-3 -3 Now solve the -3 -3 

2x < 8 inequalities. 2x > -14 

2 2 2 2 

x < 4 and 

x > -7 

{x | -7 < x < 4} 

Notice the inequality switch and 

sign change on the 11! 

Graph it: 

-8 -6 -4 -2 0 2 4 



When the problem was an absolute value “less than” something, like 

this: 

|x - #| < # 

the solution had “and” in it because the set of answers were contained 

between two numbers. 

When the problem is an absolute value “greater than” something, like 

this: 

|x - #| > # 

the solution will be different. 

The solution set is not contained by the end number. 

The solution set will be outside the bounds of the end number. 

The solution will NOT contain “and”, so it must contain… 

OR. 


3


Look at |x| > 4. 

Remember this can be written as: 

|x - 0| > 4. 

This tells us, “The distance between x and zero is greater than 4 units.” 

Graphically: 


Run four units to the right, but 

do not include the number four 

since it is not “or equal to.” 


distance is “greater than 4,” 

we need to shade everything 

to the right of 4. 

Now, run four units to the left, 

but do not include the number 

negative four since it is not “or 

equal to.” 


distance is “greater than 4,” we 

need to shade everything to the 

left of -4. 

The solution set is: 

{x | x < -4 or x > 4} 

Out of set-builder 

notation: 

x > 4 or x < -4 


-6 -4 -2 0 2 4 6 

We will solve greater than absolute value inequalities 

the same way as lesser than absolute value inequalities. 

Just remember: 

< yields an and answer, and 

> yields an or answer. 



EXAMPLE 3: Solve |5 + 2y| ≥ 3 and graph the solution set. 

Use what we just saw to rewrite the problem as two inequalities connected with “or.” 

5 + 2y ≥ 3 or 5 + 2y ≤ -3 

-5 -5 Now solve the -5 -5 

2y ≥ -2 inequalities. 2y ≤ -8 

2 2 2 2 

y ≥ -1 or 

y ≤ -4 

{y | y ≥ -1 or y ≤ -4} 

Notice the inequality switch and 

sign change on the 11! 

Graph it: 

-8 -6 -4 -2 0 2 4 


4


HOMEWORK 

Page 424 

#19 - 37 odd 


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Solving Open Sentences Involving Absolute Value 7-6

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