Presentation on theme: "Summary Subsets of Real Numbers"— Presentation transcript:
1Summary Subsets of Real Numbers EQ: How do you identify and use properties of real numbers?
2The maximum speed of a roller coaster is given by the formula s = . Roller CoasterPlay Video(1.7 MB)The maximum speed of a roller coaster is given by the formula s = .
3Summary Subsets of Real Numbers (R) Natural numbers (N) are the numbers used for counting. Whole numbers (W) are the natural numbers and 0.
4Integers 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.
5Rational NumbersRational 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
6Irrational 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.
27Solving InequalitiesEQ: What are the differences between solving equations and solving inequalities?
28VocabularyA 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.
29Solving and Graphing Inequalities Solve the inequality. Graph the solution.-3x – 12 < 3
36Real World ConnectionA 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
51Solving Absolute Value Equations EQ: How do you solve an absolute value equation?
52Definition 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| = aIf a < 0, then |a| = -aSo, the absolute value of a negative number, such as -5, is its opposite, -(-5) or 5.
611.8 Solving Absolute Value Inequalities EQ: How do you solve and graph absolute value inequalities?
62Absolute 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
63Solving Inequalities of the Form |x|≥b Solve |3x + 6|≥ 12. Graph the solution.
64Solving Inequalities of the Form |x|≥b Solve |6 – 2x | + 4 ≥ 40. Graph the solution.
65Solving Inequalities of the Form |x|<b Solve |2x – 3|< 3 . Graph the solution.
66Solving Inequalities of the Form |x|<b Solve |2x + 3| < -2. Graph the solution.
67Solving Inequalities of the Form |x|<b Solve ½ |2x + 3| < 8. Graph the solution.
68Solving Inequalities of the Form |x|<b Solve 3|2x + 6| - 9 < 15. Graph the solution.