Absolute Value Equations
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Solving Absolute Value Equations

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DEFINITION OF ABSOLUTE VALUE

For any real number a:

The symbols "– a"  should be read as "the opposite of a," not "negative a."

bulletThe absolute value of a number represents the number of units
it is from zero on the number line. 
bulletAbsolute value is a distance.  It is non-negative.
bulletWe must always consider both cases when solving absolute
value equations.

 EXAMPLE 1:


Positive Case:
Negative Case:



The solutions are {2, – 7}

Sometimes equations have extraneous solutions.


 EXAMPLE 2:

Positive Case:
Negative Case:




CHECK:


CHECK:
The only solution is w = 7.

Sometimes an equation has no solution.


 EXAMPLE 3:


Since the absolute value of a number is always zero or positive, there is no replacement for x that will make this sentence true.   The solution set has no members.  It is called the empty set.
The empty set is symbolized by Ø or {  }

Solving Absolute Value Inequalities

Defining Terminology

A sentence like " " is called a conjunction.  A conjunction of two statements is formed by connecting them with the word "and."  A conjunction is true when both statements are true.  The solution set of a conjunction is the intersection of the two graphs. Similarly, the example "" is also a conjunction.

A sentence like "x < –3 or x > 6" is called a disjunction.  A disjunction of two statements is formed by connecting them with the word "or." A disjunction is true when one or both statements are true. The solution set of a disjunction is the union of the two graphs.  

Compound Inequalities – Conjunctions

bulletTo solve a compound inequality, you must solve each part of the inequality separately.
bulletA compound inequality containing and is true only if both parts of it are true.
bulletThis means, the graph of a compound inequality containing and must be the intersection of the graphs of the two solution parts.
bulletWhere the graphs overlap or intersect determines the solution set.

 EXAMPLE 1:

 

and


 Notice the open circles around 1 and 6 indicates that these points 
are not included.

Compound Inequalities – Disjunctions

bulletTo solve a compound inequality, you must solve each part of the inequality separately.
bulletA compound inequality containing or is true only if one or more of the inequalities is true.
bulletThis means, the graph of a compound inequality containing or must be the union of the graphs of the two solution parts.

 EXAMPLE 2:

 

or


The solution set is written as:


Notice the open circles around – 5 and – 1 indicates that these points
  are not included.

Inequalities and Absolute Value

bulletTo solve an inequality of the form where b is a positive number. we solve the conjunction  
– b < A < b. A similar rule holds for .
bulletTo solve an inequality of the form where b is a positive number, we solve the disjunction  
A < – b or A > b.  A similar rule holds for .


 EXAMPLE 3:

and


The solution set is written as:


Notice the open circles around – 8 and – 4 indicate that these points are not included.

 


 EXAMPLE 4:

or

The solution set is written as:


Notice the closed circles around – 2.5 and 4 indicate that these points are included.

Extraneous Solutions

bulletSome absolute value inequalities have no solution.  For example: is never true.  Since the absolute value of a number is never negative, there is no replacement for x that will make this sentence true.  Therefore, the solution set to this inequality is the empty set.
bulletOther absolute value inequalities are always true.  One such inequality is .  The solution set of this inequality is all real numbers.  Think about the definition of absolute value.  Since any replacement for x will result in a number greater than –6, all real numbers will work.
bullet We mention these exceptions as a caution. You should test values that you believe to be contained in the solution set before making final decisions.