summary subsets of real numbers eq: how do you identify and use properties of real numbers?

Summary  Subsets of Real Numbers

EQ: How do you identify and use properties of real numbers?

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Summary  Subsets of Real Numbers (R)

 Natural numbers (N) are the numbers used for counting.

 Whole numbers (W) are the natural numbers and 0.

Integers

 The integers (Z) are the natural numbers (positive integers), zero, and the negative integers.

 Each negative integer is the opposite, or additive inverse, of a positive integer.

Rational Numbers

Rational numbers (Q) are all the numbers that can be written as quotients of integers. Each quotient must have a nonzero denominator.  

Some rational numbers can be written as terminating decimals. For example, 1/8= 0.125.

 All other rational numbers can be written as repeating decimals. For example, 1/3 = . 3

Irrational Numbers

 Irrational numbers (I) are numbers that cannot be written as quotients of integers.

 Their decimal representations neither terminate nor repeat.

 If a positive rational number is not a perfect square such as 25 or 4/9, then its square root is irrational.

 Subsets of Real Numbers

Homework

Page 6 Exercises 7-23 odd

Order of Operations

How do you use the order of operations to simplify algebraic expressions?

Get your computer!

www.mathisfun.com Click on algebra, then click on order of

operations.

Read the through the examples on the website, then take the eight question quiz.

Mistakes

There is a mistake in each of the following problems.

Discover what was done incorrectly.

-20 is correct. 2 ¼ is correct.

17 is correct.

12 4 8

8 8 64

14

9 12 3

9 36

15 3 5

15 8

7

Homework

Solving Equations

Vocabulary

A number that makes the equation true is a solution of an equation.

Solving an Equation with a Variable on Both Sides

Solve 13y + 48 = 8y − 47.

You Try

Solve 8z + 12 = 5z – 21z = -11

Solve 2t – 3 = 9 – 4tt = 2

Using the Distributive Property

Solve 3x − 7(2x − 13) = 3(−2x + 9)

You Try

Solve 6(t – 2) = 2(9 – 2t)

t = 3

Solving a Formula for One of Its Variables

Geometry: The formula for the area of a trapezoid is A = h(b1 + b2). Solve the formula for h.

Your Turn

Solve the formula for the area of a trapezoid for b1.

A = ½h(b1 + b2)

12

2bb

h

A

Solving an Equation for One of Its Variables

Solve ax +10 = bx + 3 for b.

Your Turn

Solve ax + bx – 15 = 0 for a.

x

bxa

15

Real-World Connection

Construction A dog kennel owner has 100 ft of fencing to enclose a rectangular dog run. She wants it to be 5 times as long as it is wide. Find the dimensions of the dog run.

Your Turn

A rectangle is twice as long as it is wide. Its perimeter is 48 cm. Find its dimensions.

x = 8

Homework

Worksheet

Review/Quiz

Solving Inequalities

EQ: What are the differences between solving equations and solving inequalities?

Vocabulary

A solution of an inequality in one variable is any value of the variable that makes the inequality true. Most inequalities have many solutions.

The graph of a linear inequality in one variable is the graph on the real number line of all solutions of the inequality.

Solving and Graphing Inequalities

Solve the inequality. Graph the solution.

-3x – 12 < 3

Your Turn

Solve 3x – 6 < 27. Graph the solution.

x <11

Solving and Graphing Inequalities

Solve the inequality. Graph the solution.

6 + 5(2 – x) ≤ 41

Your Turn

Solve 12 > 2(3x + 1) + 22 . Graph the solution.

x < -2

No Solutions or All Real Numbers as Solutions Solve each inequality. Graph the solution.

2x – 3 > 2(x – 5)

No Solutions or All Real Numbers as Solutions Solve each inequality. Graph the solution.

4(x – 3)+7 ≥ 4x +1

Your Turn

Solve 2x < 2(x + 1) + 3. Graph the solution.

Real World Connection

A band agrees to play for $200 plus 25% of the ticket sales. Write an inequality to model the situation. The solve the inequality to determine the ticket sales needed for the band to receive at least $500.

Homework

Page 41 Exercises 11, 14, 15-18, 21-26, 41

Solving Compound Inequalities

EQ: How do you solve and graph a compound sentences and inequalities

using and and or?

Vocabulary

A compound inequality is a pair of inequalities joined by and or or.

To solve a compound inequality containing and, find all values of the variable that make both inequalities true.

Compound Inequality Containing And

Graph the solution of

3x − 1 > −28 and 2x + 7 < 19.

Compound Inequality Containing And

Graph the solution of

-2x + 3 < 4x +2 and -2x +3> -4x - 2

Compound Inequality Containing And

Graph the solution of 5 < -2x + 1/2 < 27/3.

Your Turn

Graph the solution of 2x > x + 6 and x – 7 < 2.

Compound Inequality Containing Or

Graph the solution of

4y − 2 ≥ 14 or 3y − 4 ≤ −13.

Compound Inequality Containing Or

Graph the solution of 3x + 1 < 7 or 2x - 9 > 7.

Compound Inequality Containing Or

Graph the solution of

-3 ≤ 2y + 9 or 18 > 4y – 10.

Your Turn

Solve x – 1 < 3 or x + 3 > 8. Graph the solution.

Ticket Out

Graph given the solution set is {x|x > 4 and x > 2},

then graph given the solution set is

{x|x > 4 or x > 2} .

Compare the two graphs, what is your conclusion?

Homework

x + 7 > -2 or x - 4 < 8

Page 41 Exercises 27-38

Day 2

Homework: Page 41-42 Exercises 12, 13, 19-20, 27-38,

40.

Solving Absolute Value Equations

EQ: How do you solve an absolute value equation?

Definition of Absolute Value

The absolute value of a number is its distance from zero on the number line and distance is nonnegative.

For any real number a: If a ≥ 0, then |a| = a If a < 0, then |a| = -a

So, the absolute value of a negative number, such as -5, is its opposite, -(-5) or 5.

Solving Multi-Step Absolute Value Equations

Solve |4x – 1| – 5 = 10

Example

Solve |x – 3 | + 2 = 4

Example

622

3 Solve

x

No Solution

Solve |2x + 3| +10 = 5

Examples

Solve -2|x – 3| = 12

Solve -5|x +1|= -8

Your Turn

1) |3x + 2| = 7

2) 2|3x – 1| + 5 = 33

Work in Pairs

Write an absolute value equation for your partner to solve, then check one another’s work.

Homework

Page 47 Exercises 5-18

1.8 Solving Absolute Value Inequalities

EQ: How do you solve and graph absolute value

inequalities?

Absolute Value Inequalities

Let k represent a positive real number.

|x| ≥ k is equivalent to x ≤ -k or x ≥ k.

|x|≤ k is equivalent to x ≤ k and x ≥- k

(-k ≤ x ≤ k)

Hint: greatOR -- greater is an or

less thAND-- less than is an and

Solving Inequalities of the Form |x|≥b

Solve |3x + 6|≥ 12. Graph the solution.

Solving Inequalities of the Form |x|≥b

Solve |6 – 2x | + 4 ≥ 40. Graph the solution.

Solving Inequalities of the Form |x|<b

Solve |2x – 3|< 3 . Graph the solution.

Solving Inequalities of the Form |x|<b

Solve |2x + 3| < -2. Graph the solution.

Solving Inequalities of the Form |x|<b

Solve ½ |2x + 3| < 8. Graph the solution.

Solving Inequalities of the Form |x|<b

Solve 3|2x + 6| - 9 < 15. Graph the solution.

Homework

Page 47

Exercises 19-34

Review/Test