 # Summary Subsets of Real Numbers

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Summary Subsets of Real Numbers
EQ: How do you identify and use properties of real numbers?

The maximum speed of a roller coaster is given by the formula s = .
Roller Coaster Play Video (1.7 MB) The maximum speed of a roller coaster is given by the formula s =       .

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=  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?

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.

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 – 21 z = -11 Solve 2t – 3 = 9 – 4t t = 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)

Solving an Equation for One of Its Variables
Solve ax +10 = bx + 3 for b.

Your Turn Solve ax + bx – 15 = 0 for a.

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) 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. \$200+25%>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 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

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