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Absolute Value Equations. Objective. I will be able to solve absolute value equations. -2. -1. 0. 1. 2. 3. Absolute Value. Absolute value of a number is its distance from zero on a number line. 2 units. Solving Equations of the Form. Example 1. Solve - PowerPoint PPT Presentation
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Absolute Value Equations
Objective I will be able to solve
absolute value equations.
Absolute Value
Absolute value of a number is its distance from zero on a number line.
-2 20 1-1 3
2 units
2 2
Solving Equations of the Form
,If a is a positivenumber then X a
isequivalent to X aor X a
X a
Example 1
Solve Since 2 is positive is equivalent to p = 2
or p = -2 To check, let p = 2 and then p = -2 in the
original equation.
2p p
2p Original Equation
2 2
2 = 2
Let p = 2
True
2p Original Equation
2 2 Let p = -2
2 = 2
True
Solution
The solutions are 2 and -2 or the solution set is {2, -2}
Give it a try!
Solve 5y
Example 2
Solve
Translate: 5 w + 3 = 7 OR 5 w + 3 = -7
Solve both equations for w
5 w + 3 = 7 5 w + 3 = -7
5w = 4 5 w = -10
w = w = -2
5 3 7w
45
Solution
{-2, }
Check your solution, let w = -2 then let w = -4/5
45
Give it a try!
Solve 4 2 6x
Example 3
Solve 1 112x
24 and -20
Give it a try!
Solve 4 13x
Example 4: Isolate the absolute value expression! Solve
2 5 7x
2 5 7x Subtract 5 from both sides
2 2x
2x = 2 2x = -2x = 1 x = -1
The solutions are -1 and 1
Give it a try!
Solve 5 5 7x
Example 5: ZERO
Solve
We are looking for all numbers whose distance from 0 is zero units. The only number is 0. The solution is 0.
0y
Example 6
Solve: 2 25 23x
2 2x
1x
Subtract 25 from both sides
Divide both sides by 2
The absolute value of a number is NEVER negative, so this equation has no solution!
Give it a try!
Solve: 3 12 6y
Example 7
Solve: 3 12
2
x
The absolute value of any expression is never negative, so no solution exists!
When are absolute value expressions equal?
2 2
2 2
2 2
2 2
Same
Same
Opposites
Opposites
Two absolute value expressions are equal when the expressions inside the absolute value bars are equal to or are opposites of teach other.
Example 8:
Solve:
This equation is true if the expressions inside the absolute value bars are equal to or are opposites of each other.
3x + 2 = 5x – 8 OR 3x + 2 = -(5x -8)
3 2 5 8x x
Solve each equation
3x + 2 = 5x – 8 OR 3x + 2 = -(5x – 8)
-2x + 2 = -8 3x + 2 = -5x + 8
-2x = 10 8 x = 6
x = 5 x = ¾
The solutions are ¾ and 5.
Give it a try!
Solve: 4 5 3 5x x
Example 9
Solve:
x – 3 = 5-x OR x – 3 = -(5-x)
2 x – 3 = 5 x – 3 = -5 + x 2x = 8 x-x – 3 = - 5
x = 4 0 – 3 = -5
-3 = -5
False
3 5x x
Solution to Example 9
The equation on the right simplified to a false statement. So the only solution to this equation is 4.
Give it a try!
Solve 2 4x x