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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

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drill 4
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
Drill #9

Solve 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
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
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 4 solving absolute value equations review of major points
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
1-5 Solving Inequalities

Objective: To solve and graph the solutions to linear inequalities.

trichotomy property
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
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
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
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
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 > 36

Ex2: 17 – 3w > 35

Ex3:

set builder notation
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
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
Special Cases

Solve the following inequalities and graph their solutions on a number line.

Ex1:3(1 – 2x) < -6(x – 1)

Ex2:

writing inequalities from verbal expressions
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