Drill #10

Solve the following absolute value equalities. Remember to solve for both cases (positive and negative) and check your answers.

1. |2x – 3| = 12

2. |5 + x| + 2 = 2

3. 3 |2x – 1| + 6 = – 3

Drill #11

Solve the following absolute value equalities. Remember to solve for both cases (positive and negative) and check your answers.

1. |x + 4| = 20

2. |2x – 6| + 2 = 2

3. 3 |2x – 1| + 1 = 10

1-6 Solving Inequalities

Objective: To solve inequalities and graph the solution sets.

Calorie Intake

1. Multiply your body weight w by 4.3

2. Multiply your height h by 4.7

3. Add the numbers

4. Add 655 to your result from step 3

5. Multiply your age a times 4.7

6. Subtract the product in step 5 from the expression in step 4

Maintaining body weightThe expression for optimal Calorie intake is:

4.3w + 4.7h + 655 – 4.7a

Multiply by 1.3 to find optimal intake with moderate activity

1.3 (4.3w + 4.7h + 655 – 4.7a)

How many calories do you need to consume to maintain your weight?

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.

(1.) 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

(2.) 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

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

(3.) 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

(4.) Solving Inequalities*• solve inequalities the same way as equations

(using S. G. I. R.)EXCEPTIONS:

Change the inequality sign when you:– multiply or divide by a negative number. – swap sides (non-symmetry property)

Example #1*: -4x + 6 > 10Example #2*: 3x > 4x + 2 – x Write your solution in a solution set.

Set Builder Notation**

Definition: The solution x < -1 written in set-builder notation:

{x| x < -1}

We say, the set of all x, such that x is less than -1.

Empty Set**

Definition: The set having no members, symbolized by { } or O

When an equation has no solution, the answer is said to be null or the empty set.

Classwork

1-6 Study Guide

#1-4

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)

Example #1*: Graph the solution to the last example

Classwork

1-5 Practice

#1-2

Writing Inequalities (#11)*

Define a variable and write an inequality for each problem then solve and graph the solution:

4. The product of 11 and a number is less than 53.

5. The opposite of five times a number is less than 321.

Test Scores

Ron’s score on the 1st three of four 100-point chemistry tests were 90, 96, and 86. What must he score on his fourth test to have an average of at least 92 for all the tests?