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Drill #4. Evaluate 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. Drill #9. Solve the following equations: Check your solutions! 1. 2|x – 2| + 3 = 3 - PowerPoint PPT Presentation
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Drill #4
Evaluate 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
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
Drill #13
Solve the following absolute value equalities:
1. |3x – 3| = 12
2. 2|5 – x | + 4 = 2
3. -3|x – 1| + 1 = 1
4. |2x – 1| = 4x – 7
Drill #14
Solve the following absolute value equalities:
1. |3x + 8| = -x
2. 2|5 – x | – 3 = – 3
Solve the following inequalities and graph their solutions on a number line:
3. 12 – 3x > 16
4. 1
7
23
x
1-4 Solving Absolute Value Equations: Review of major points
• -isolate the absolute value (if its equal to a neg, no solutions)
• -set up two cases (the absolute value is removed)
• -solve each case.
• -check each solution. (there can be 0, 1, or 2 solutions)
1-5 Solving Inequalities
Objective: To solve and graph the solutions to linear inequalities.
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.
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
Multiplication and Division Properties for Inequalities
For positive numbers:
1. If c > 0 and a < b then ac < bc and a/c < b/c
2. If c > 0 and a > b then ac > bc and a/c > b/c
NOTE: The inequality stays the same
For negative numbers:
3. If c < 0 and a < b then ac > bc and a/c > b/c
4. If c < 0 and a > b then ac < bc and a/c < b/c
NOTE: The inequality changes
Example: -2x > 6
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
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.
Ex1: 2x + 4 > 36Ex2: 17 – 3w > 35
Ex3: 4)5(3
1 x
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.
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)
Special Cases
Solve the following inequalities and graph their solutions on a number line.
Ex1: 3(1 – 2x) < -6(x – 1)
Ex2: 12
1
4
62
x
x
Writing Inequalities from Verbal Expressions
Define a variable, and write an inequality for each problem, then solve and graph the solution.
Ex1: Twelve less than the product of three and a number is less than 21.
Ex2: The quotient of three times a number and 4 is at least -16
Ex3: The difference of 5 times a number and 6 is greater than the number.
Ex4: The quotient of the sum of a number and 6 is less than -2