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