Absolute Value Equations
Absolute Value
There are three definitions of absolute value:
Arithmetic Definition:
The absolute value of a real number n is the maximum of the number n and
its additive inverse (­n). In symbols:n
|n| = n max (­n)
Algebraic Definition:
The absolute value of a real number n is defined as
n if n ≥ 0
|n| = ­n if n < 0
Geometric Definition:
The absolute value of a real number n is the distance between the graphs of the numbers n and 0 on the real number line.
{
How do we interpret:
|x| = b
or |x - 0| = b
What numbers are a distance of b units from 0?
Find the number, x, such that the difference between x and 0 is b.
How do we interpret:
|x - a| = b
What numbers are a distance of b units from a?
Find the number, x, such that the difference between the number x and a is b.
|x - a| = b
center
bound ­ the number above/below the center
Write an absolute value equation with solutions:
4, 6
|x - 5| = 1
­4, 8
62", 74"
|x - 2| = 6
|x - 68"| = 6"
­6, ­2
|x + 4| = 2
Absolute Value Equations
Absolute Value
Solve the following absolute value equations (Make sure you check):
Write two equations:
Solve:
Check:
Write your solution:
Isolate the absolute value:
|y + 9| = 21
or y + 9 = 21 or
y = 12
|12 + 9| = 21 √
{­30, 12}
|a - 3| - 14 = ­6
y + 9 = ­21
y = ­30
|­30 + 9| = 21 √
|a - 3| = 8
Write two equations:
Solve:
Check:
Write your solution:
or a - 3 = 8 or
a = 11
|11 - 3| = 8 √
4|3t + 8| = 16t{­5, 11}
Isolate the absolute value:
Write two equations:
Solve:
Check:
Write your solution:
|3t + 8| = 4t
3t + 8 = 4t
or or
t = 8
4|32| = 16(8) √
{8}
a - 3 = ­8
a = ­5
|­5 - 3| = 8 √
3t + 8 = ­4t
t = ­8/7
does not √
When you write the two equations, the first equation is the original equation without the absolute value symbols. For the second equation write an equation without the absolute value symbols where:
1. the absolute value expression is negated, or
2. the terms originally outside of the absolute value expression are negated (Do not negate terms being multiplied or divided by the absolute value expression).
Absolute Value Equations
Absolute Value
Solve for p and check.
3|p - 5| = 2p
Method 1:
3(p - 5) = 2p or 3(­p + 5) = 2p
Method 2:
3(p - 5) = 2p or 3(p - 5) = ­2p
3p - 15 = 2p or ­3p + 15 = 2p
3p - 15 = 2p or 3p - 15 = ­2p
p - 15 = 0 or 15 = 5p
p - 15 = 0 or ­15 = ­5p
p = 15 or p = 3
p = 15 or p = 3
Check
3|p - 5| = 2p
3|p - 5| = 2p
3|15 - 5| = 2(15)
3|3 - 5| = 2(3)
30 = 30 √
Solution: {3, 15}
6 = 6 √