Download Algebra_1A_files/Lecture 2-2

Ch 2.2
Objective:
To solve problems involving absolute
value of numbers or variables.
Definitions
Opposite (-)
Take the negative of a value to create the opposite of that number.
For example: The opposite of 4 is -4. The opposite of -4 is –(-4) = 4.
Absolute Value (| |)
Represents the distance (which is always positive) from 0 on a number line.
For example: The absolute value of -2 is 2 which is written as |-2| = 2
Solving Absolute Value Problems
If you remove the absolute value sign, you must replace it with BOTH a positive sign AND
a negative sign.
For example: Solve |x| = 2
This results in two equations: +x = 2 and –x = 2
Try These
• What is the opposite of 0? ______
• What is the opposite of 10? _____
• What is the opposite of -6? ______
•
•
•
•
What is the absolute value of 0 written as |0|? ________
What is the absolute value of 7 written as |7|? ________
What is the absolute value of -5 written as |-5|? _______
What is the opposite of the absolute value of -2
written as -|-2|? _________
Absolute value:
The distance from
zero on the number line.
-5 -4 -3 -2 -1
0
1 2
3
4
5
3 3
12  12
5  5
 7  -7
10  10
 23  -23  16  -16
0 0
5.4  5.4
Opposites vs. Absolute Value
Given Number
Opposite
Absolute Value
8
-8
8 8
-24
+24
24  24
-3.5
+3.5
35
.  35
.
1
4
2
1
4
2
1
1
4 4
2
2
Solve each equation below.
1) x  10
x = 10 or -10
2) x  4
x = 4 or - 4
3) x  0
x=0
4) x  5
“no solution”
5)  x  14
x = 14 or - 14
3
6) t 
4
3
3
t = 4 or - 4
Determine whether each statement is true always,
sometimes, or never for all real numbers.
1) x  x
sometimes
2)  x  x
always
3) x   x
sometimes
4) x  0
never
5)  x   x
sometimes
6) x   x
always
Velocity vs. Speed
Velocity - Indicates speed and direction.
Speed - The absolute value of velocity.
Example:
A helicopter descends at 50 feet/second.
A) What is its velocity?
B) What is its speed?
-50 ft./sec.
+50 ft./sec.
Counterexamples
To prove a statement true, it must be
proven true for all examples - difficult!
Counterexample - An example that proves
a statement false.
Statement: All pets are furry.
Counterexample: Goldfish.
Statement:
xx
Counterexample: x  0
Determine whether each statement is true or false
for all real numbers. If it is false, find a counterexample that proves it is false.
1) x  0
4) x   x
False, x = 0
2) x  x
False, x = -7
3)  x   x
False, x = -5
True
5) x   x
True
6) x  x
True