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Drill #4Evaluate the following if a = -2 and b = ½. 1. ab – | a – b |
Solve the following absolute value equalities: 2. |2x – 3| = 12
3. |5 – x | + 4 = 2
4. |x – 2| = 2x – 7
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Drill #9Solve the following equations:Check your solutions!1. 2|x – 2| + 3 = 3
2. -3|2x + 4| + 2 = –1
3. |2x + 2| = 4x + 10
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Writing Absolute Value Equations from Word Problems*
• Find the value that is + or –. This will be the value that is on the opposite side of the abs. val.
• Find the middle value. This value will be subtracted from the variable in the abs. val.
Example: expected grade = 90 +/- 5 points
this translates to |x – 90| = 5where x = test grade
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1-5 Solving Inequalities
Objective: To solve and graph the solutions to linear inequalities.
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Trichotomy Property
Definition: For any two real numbers, a and b, exactly one of the following statements is true:
a < b a = b a > b
A number must be either less than, equal to, or greater than another number.
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Addition and Subtraction Properties For Inequalities
1. If a > b, then a + c > b + c and a – c > b – c
2. If a < b, then a + c < b + c and a – c < b – c
Note: The inequality sign does not change when you add or subtract a number from a side
Example: x + 5 > 7
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Multiplication and Division Properties for Inequalities
For positive numbers:1. If c > 0 and a < b then ac < bc and a/c < b/c2. If c > 0 and a > b then ac > bc and a/c > b/cNOTE: The inequality stays the same
For negative numbers:3. If c < 0 and a < b then ac > bc and a/c > b/c4. If c < 0 and a > b then ac < bc and a/c < b/cNOTE: The inequality changes
Example: -2x > 6
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Non-Symmetry of Inequalities
If x > y then y < x
• In equalities we can swap the sides of our equations:
x = 10, 10 = x
• With inequalities when we swap sides we have to swap signs as well:
x > 10, 10 < x
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Solving Inequalities• We solve inequalities the same way as
equalitions, using S. G. I. R.• The inequality doesn’t change unless we
multiply or divide by a negative number.
Example #1*: Single StepEx1: y – 6 < 3Ex2: 5w + 3 > 4w + 9Ex3: 5x – 3 > 4x + 2
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Set Builder Notation
Definition: The solution x > 5 written in set-builder notation:
{x| x > 5}
We say, the set x, such that x is greater than 5.
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Graphing inequalities• Graph one variable inequalities on a number
line.• < and > get open circles • < and > get closed circles• For > and > the graph goes to the right. (if the variable is on the left-hand side)• For < and < the graph goes to the left. (if the variable is on the left-hand side)
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Solve Multi-Step Inequalities: Examples
(On board)