Geometric Definition of Absolute Value
|x| is defined geometrically as the distance from x to the origin.
Looking at an example: |-4| and |4|
~ Geometrically, if we were to use a number line and count the number of steps from -4 to 0, it would be 4.
~ If we were to use a number line to count the number of steps from 4 to 0, it would 4.
~ So, we would say that the absolute value of 4 or -4 would be 4 in both cases.
Algebraic Definition of Absolute Value
|x| is defined algebraically as an equation.
Looking at an example: |x| = 3
~ We can assume that x can be either -3 OR it can be +3.
Solving an Absolute Value Equation
Remember, you are finding the solution(s) to the equation.
Notice the word OR is emphasized in the above example.
This type of equation is known as a disjunction - OR must be included in the setup.
The setup is as follows:
|x| = x when x ≥ 0 OR |x| = -x when x < 0
Example 1: If |x| = 3, find the value(s) of x.
Solution: According to the algebraic definition of absolute value.
x = 3 OR -x = 3
x = 3 OR x = -3
Example 2: If | x + 2| = 5, find the value(s) of x.
Solution: According to the algebraic definition of absolute value.
(x + 2) = 5 OR - (x + 2) = 5
(x + 2) = 5 OR (x + 2) = -5
Let's do the algebra...
x = 3 OR x = -7
Absolute Values with a Given Condition
We will look at solving absolute value expressions with a given condition.
Example: Simplify the expression |x - 3| given that x < 3
.
Steps to solve these problems
~ Look at the given condition.
~ In the above example - substitute x with a value less than (but not equal to) 3.
~ Determine if the result of the substitution has a positive or a negative value.
Solution
1. Problem | x - 3 | given that x < 3
2. Replace x with a value less than 3, use 2 |2 - 3| = -1, the result is negative.
3. Include a negative sign before the expression -(x - 3)
4. Simplify -x + 3
Let's use the same absolute value expression, but this time the condition is x > 3.
Solution
1. Problem | x - 3 | given that x > 3
2. Replace x with a value less than 3, use 5 |5 - 3| = 2, the result is positive
3. Include a negative sign before the expression +(x - 3)
4. Simplify x - 3
Another absolute value expression might include two (or more) absolute value expressions.
Example: Simplify the expression |x - 5| + |x - 4| given that 4 < x < 5.
Solution
1. Problem |x - 5| + |x - 4| given that 4 < x < 5.
2. Replace x with 4.5 |4.5 - 5| + |4.5 - 4| will result in a -.5
for the first expression and a .5 for the second absolute value expression.
3. Rewrite as an algebraic expression -(x - 5) + (x - 4)
4. Simplify -x + 5 + x - 4 = 1