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Absolute Value Equations

# Absolute Value Equations

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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