Summary
Summary IEB AP Math:Absolute Values
This document describes: -How to solve absolute value equations -How to solve absolute value graphs
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Absolute values:Similar to distance:
It’s the distance of x from zero
Example:
𝑥 = −1 𝑥=0 𝑥=1
So:
|𝑥| = 𝑥
|−1| = 1
|1| = 1
Absolute values is defined as:
𝑥 𝑖𝑓 𝑥 ≥ 0
𝑓(𝑥) = {
−𝑥 𝑖𝑓 𝑥 < 0
The x is the elements between the ||
Example:
𝑥 − 1 𝑖𝑓 𝑥 − 1 ≥ 0
|𝑥 − 1| = {
−(𝑥 − 1)𝑖𝑓 𝑥 − 1 < 0
2𝑥 + 2 𝑖𝑓 2𝑥 + 2 ≥ 0
|2𝑥 + 2| = {
−(2𝑥 + 2)𝑖𝑓 2𝑥 + 2 < 0
Statement:
Solve for:
|𝑥 − 2| − 2𝑥 = 0
There are two cases when 𝑥 − 2 ≥ 0 𝐴𝑁𝐷 𝑥 − 2 < 0
, Case 1:
𝑥−2≥0
𝑥≥2
Thus:
𝑥 − 2 − 2𝑥 = 0
−𝑥 = 2 ( 𝑥 = −2)
However, since our parameters is that 𝑥 ≥ 2, this answer is invalid
Case 1:
𝑥−2<0
𝑥<2
Thus:
−(𝑥 − 2) − 2𝑥 = 0
−𝑥 − 2𝑥 + 2 = 0
−3𝑥 = −2
2
𝑥=
3
However, since our parameters is that 𝑥 < 2, this answer is valid.
Sub in the values into the orginal statement to see if RHS = LHS
Solve for:
|𝑥 − 3| = 5
Lets get our parameters:
𝑥 𝑖𝑓 𝑥 ≥ 3
−𝑥 𝑖𝑓 𝑥 < 3
For case 1:
𝑥−3=5
𝑥=2
Which is not 𝑥 ≥ 3, thus invalid answer
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