Absolute Value: Definition, Properties, and Examples, Slides of Algebra

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§4.3a Absolute

Value

Review §

 Any QUESTIONS About

• §4.2 → InEqualities & Problem-Solving

 Any QUESTIONS About HomeWork

• §4.2 → HW-

4.2 MTH 55

Graph y = | x |

• Make T-table x y = | x | -6 6 -5 5 -4 4 -3 3 -2 2 -1 1 0 0 1 1 2 2 3 3 4 4 5 5 6 6

x

y

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0

1

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3

4

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6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls

Absolute Value Properties

1. | ab | = | a |· | b | for any real numbers a & b
   • The absolute value of a product is the product of the absolute values
2. | a / b | = | a |/| b | for any real numbers a & b ≠ 0
   • The absolute value of a quotient is the quotient of the absolute values
3. |−a| = |a| for any real number a
   • The absolute value of the opposite of a number is the same as the absolute value of the number

Distance & Absolute-Value

• For any real numbers a and b , the distance between them is | a – b |
• Example  Find the distance between − 12 and −56 on the number line
• SOLUTION
  • |− 12 − (−56)| = |+44| = 44
  • Or
  • |− 56 − (−12)| = |−44| = 44

Example  AbsVal Expressions

• Find the Solution-Sets for

a) | x | = 6 b) | x | = 0 c) | x | = −

• SOLUTION a) | x | = 6
• We interpret | x | = 6 to mean that the number x is 6 units from zero on a number line.
• Thus the solution set is {−6, 6}

Example  AbsVal Expressions

• Find the Solution-Sets for

a) | x | = 6 b) | x | = 0 c) | x | = −

• SOLUTION c) | x | = −
• Since distance is always NonNegative, | x | = − has NO solution.
• Thus the solution set is Ø

Absolute Value Principle

• For any positive number p and any

algebraic expression X :

a. The solutions of | X | = p are those numbers that satisfy X = − p or X = p

b. The equation | X | = 0 is equivalent to the equation X = 0

c. The equation | X | = − p has no solution.

Example  AbsVal Principle

• Solve: a) |2 x +1| = 5; b) |3 − 4 x | = −
• SOLUTION b) |3 − 4 x | = −
  • The absolute-value principle reminds us that absolute value is always nonnegative.
  • So the equation |3 − 4 x | = −10 has NO solution.
  • Thus The solution set is Ø

Example  AbsVal Principle

• Solve |2 x + 3| = 5
• SOLUTION
  • For |2 x + 3| to equal 5, 2 x + 3 must be 5 units from 0 on the no. line. This can happen only when 2 x + 3 = 5 or 2 x + 3 = −5.
  • Solve Equation Set

2 x + 3 = 5 or 2 x + 3 = – 2 x = 2 x = 1

2 x = – x = –

or or

• Graphing the Solutions –5 –4 –3 –2 –1 0 1 2 3 4 5

Two AbsVal Expression Eqns

• Sometimes an equation has TWO absolute- value expressions.
• Consider | a | = | b |. This means that a and b are the same distance from zero.
• If a and b are the same distance from zero, then either they are the same number or they are opposites.

Example  2 AbsVal Expressions

• Solve: |3 x – 5| = |8 + 4 x |.
• SOLUTION
  • Recall that if | a | = | b | then either they are the same or they are opposites

3 x – 5 = 8 + 4 x

This assumes these numbers are the same

This assumes these numbers are opposites.

3 x – 5 = –(8 + 4 x )

OR

 Need to solve Both Eqns for x

Solve Eqns of Form | ax + b | = | cx + d |

• To solve an equation in the form

| ax + b | = | cx + d |

1. Separate the absolute value equation

into two equations: ax + b = cx + d and

ax + b = − ( cx + d ).

2. Solve both equations.

Inequalities &AbsVal Expressions

• Example  Solve: |x| < 3 Then graph
• SOLUTION
  • The solutions of | x | < 3 are all numbers whose distance from zero is less than 3. By substituting we find that numbers such as −2, −1, −1/2, 0, 1/3, 1, and 2 are all solutions.
  • The solution set is { x | −3 < x < 3}. In interval notation, the solution set is (−3, 3). The graph:

-3 3

( (^) )