summary subsets of real numbers eq: how do you identify and use properties of real numbers?
TRANSCRIPT
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Summary Subsets of Real Numbers
EQ: How do you identify and use properties of real numbers?
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Roller Coaster Play Video (1.7 MB)
The maximum speed of a roller coaster is given by the formula
s = .
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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.
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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.
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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
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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.
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Subsets of Real Numbers
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Homework
Page 6 Exercises 7-23 odd
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Order of Operations
How do you use the order of operations to simplify algebraic expressions?
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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.
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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
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Homework
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Solving Equations
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Vocabulary
A number that makes the equation true is a solution of an equation.
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Solving an Equation with a Variable on Both Sides
Solve 13y + 48 = 8y − 47.
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You Try
Solve 8z + 12 = 5z – 21z = -11
Solve 2t – 3 = 9 – 4tt = 2
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Using the Distributive Property
Solve 3x − 7(2x − 13) = 3(−2x + 9)
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You Try
Solve 6(t – 2) = 2(9 – 2t)
t = 3
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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.
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Your Turn
Solve the formula for the area of a trapezoid for b1.
A = ½h(b1 + b2)
12
2bb
h
A
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Solving an Equation for One of Its Variables
Solve ax +10 = bx + 3 for b.
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Your Turn
Solve ax + bx – 15 = 0 for a.
x
bxa
15
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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.
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Your Turn
A rectangle is twice as long as it is wide. Its perimeter is 48 cm. Find its dimensions.
x = 8
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Homework
Worksheet
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Review/Quiz
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Solving Inequalities
EQ: What are the differences between solving equations and solving inequalities?
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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.
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Solving and Graphing Inequalities
Solve the inequality. Graph the solution.
-3x – 12 < 3
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Your Turn
Solve 3x – 6 < 27. Graph the solution.
x <11
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Solving and Graphing Inequalities
Solve the inequality. Graph the solution.
6 + 5(2 – x) ≤ 41
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Your Turn
Solve 12 > 2(3x + 1) + 22 . Graph the solution.
x < -2
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No Solutions or All Real Numbers as Solutions Solve each inequality. Graph the solution.
2x – 3 > 2(x – 5)
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No Solutions or All Real Numbers as Solutions Solve each inequality. Graph the solution.
4(x – 3)+7 ≥ 4x +1
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Your Turn
Solve 2x < 2(x + 1) + 3. Graph the solution.
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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.
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Homework
Page 41 Exercises 11, 14, 15-18, 21-26, 41
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Solving Compound Inequalities
EQ: How do you solve and graph a compound sentences and inequalities
using and and or?
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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.
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Compound Inequality Containing And
Graph the solution of
3x − 1 > −28 and 2x + 7 < 19.
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Compound Inequality Containing And
Graph the solution of
-2x + 3 < 4x +2 and -2x +3> -4x - 2
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Compound Inequality Containing And
Graph the solution of 5 < -2x + 1/2 < 27/3.
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Your Turn
Graph the solution of 2x > x + 6 and x – 7 < 2.
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Compound Inequality Containing Or
Graph the solution of
4y − 2 ≥ 14 or 3y − 4 ≤ −13.
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Compound Inequality Containing Or
Graph the solution of 3x + 1 < 7 or 2x - 9 > 7.
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Compound Inequality Containing Or
Graph the solution of
-3 ≤ 2y + 9 or 18 > 4y – 10.
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Your Turn
Solve x – 1 < 3 or x + 3 > 8. Graph the solution.
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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?
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Homework
x + 7 > -2 or x - 4 < 8
Page 41 Exercises 27-38
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Day 2
Homework: Page 41-42 Exercises 12, 13, 19-20, 27-38,
40.
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Solving Absolute Value Equations
EQ: How do you solve an absolute value equation?
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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.
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Solving Multi-Step Absolute Value Equations
Solve |4x – 1| – 5 = 10
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Example
Solve |x – 3 | + 2 = 4
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Example
622
3 Solve
x
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No Solution
Solve |2x + 3| +10 = 5
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Examples
Solve -2|x – 3| = 12
Solve -5|x +1|= -8
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Your Turn
1) |3x + 2| = 7
2) 2|3x – 1| + 5 = 33
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Work in Pairs
Write an absolute value equation for your partner to solve, then check one another’s work.
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Homework
Page 47 Exercises 5-18
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1.8 Solving Absolute Value Inequalities
EQ: How do you solve and graph absolute value
inequalities?
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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
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Solving Inequalities of the Form |x|≥b
Solve |3x + 6|≥ 12. Graph the solution.
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Solving Inequalities of the Form |x|≥b
Solve |6 – 2x | + 4 ≥ 40. Graph the solution.
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Solving Inequalities of the Form |x|<b
Solve |2x – 3|< 3 . Graph the solution.
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Solving Inequalities of the Form |x|<b
Solve |2x + 3| < -2. Graph the solution.
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Solving Inequalities of the Form |x|<b
Solve ½ |2x + 3| < 8. Graph the solution.
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Solving Inequalities of the Form |x|<b
Solve 3|2x + 6| - 9 < 15. Graph the solution.
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Homework
Page 47
Exercises 19-34
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Review/Test