4.6
Absolute-Value Equations and Inequalities
Homework: EOO 1-61
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To solve equations and inequalities that contain an
expression in absolute value, we need the following relationships between the
expression in absolute value and the constant on the other side of the
equation:
(Please note that the absolute-value expression must be isolated).
A. If, then statement for equations containing absolute-value.
Example:
Example:
Example:
B. If, then statement for less than inequalities containing absolute-value.
Example:
Example:
C. If, then statement for greater than inequalities containing absolute-value.
Example:
The solution set written in interval notation is:
Example:
In all of the examples above, the statements are conditional, that is they are true statements depending on the value that is substituted for the variable. When solving conditional statements, we find the value for x, the variable, that will make the statement true.
We can have absolute-value statements that are always true for every value of x and statements that are never true for any value of x. Some examples of these types of statements follow.
The following statements are always true for every value of x. These statements have infinite solutions.
The following statements are never true for any value of x. These statements have no solution.
