# Solving and Graphing Inequalities

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1 Algebra I Pd Basic Inequalities 3A What is the answer to the following questions? Solving and Graphing Inequalities We know 4 is greater than 3, so is 5, so is 6, so is 7 and 3.1 works also. So does 3.01 and So would 3.12 and 3.13 you start to get the picture that it would be impossible to list all the possible solutions (all the possible numbers that make the inequality true ). Since we cannot list all the answers, we express the solution set by graphing on a number line. When finding the solution for an equation we get one answer for x. For 3x 5 = 16, the only solution is x = 7. When we have an inequality to solve, we have a range of numbers that can be a solution. In that range there is an infinite amount of possible numbers that make the inequality true. Before we begin our example problems, let s refresh on what each inequality symbol means: It is helpful to remember, the open part of the symbol always faces the larger quantity. Symbols Meaning Greater than Less than Greater than or equal to Less than or equal to How to Graph Basic Inequalities Steps to graph an inequality: 1) create a number line 2) place five numbers on the number line (the solution number, two numbers to the left and two numbers to the right) 3) place a circle on the solution (in this case, 3) > and < get an open circle > and < get a closed circle 4) shade to the side of the number line that contains the solution set (shade all the way to the end) Shade: Less than (and < ) Left (think L, L) Greater than (and > ) Right Examples of graphing an inequality: Part 1 Graph the inequalities on number lines: 1) 2) 3) 4)

2 Part 2 Write an inequality that represents each verbal expression: 5) v is greater than 6. 6) b is less than or equal to -1. Part 3 Write each inequality in words: 7) 8) Part 4 Solve and graph the following inequalities: Ex) 10) Check: (always check ZERO in original!!) True! If true the zero should be shaded, and if false the zero should NOT be shaded. 11) 12)

3 Algebra I Pd Basic Inequalities HW 3B Solving and Graphing Inequalities Part 1 Directions: Answer each question. All work for #1 8 can be completed on this paper. 1) A student will study French for at least 3 years. Which inequality describes the situation? (a) (b) (c) (d) 2) A child must be under 46 inches to ride the Jungle Jam. Which inequality represents the situation? (a) (b) (c) (d) 3) A child must be at most 38 inches to ride Stargazer. Which inequality represents the situation? (a) (b) (c) (d) 4) Write an inequality that represents: is greater than or equal to -2 5) Write an inequality that represents: is less than -7 6) Write an inequality that represents: 2 more than is greater than -3 7) Write an inequality that represents: 6 less than is greater than or equal to 5 8) Write an inequality for the graph: (a) (b) Part 2 Directions: Solve and graph each inequality in your homework section of your notebooks. 9) 10) 11) 12) 13) 14)

4 Algebra I Pd Basic Inequalities HW 3B HW ANSWERS 1) at least 3 years = 3 years or more 2) under 46 years = less than 46 (d) (a) 3) at most = less than or equal to (b) 4) 5) 6) 7) 8) (a) (b) 9) 10) 11) 12) 13) 14)

5 Algebra I Pd Basic Inequalities 3C What is the answer to the following questions? Solving and Graphing Inequalities (Day 2) We know 4 is greater than 3, so is 5, so is 6, so is 7 and 3.1 works also. So does 3.01 and So would 3.12 and 3.13 you start to get the picture that it would be impossible to list all the possible solutions (all the possible numbers that make the inequality true ). Since we cannot list all the answers, we express the solution set by graphing on a number line. When finding the solution for an equation we get one answer for x. For 3x 5 = 16, the only solution is x = 7. When we have an inequality to solve, we have a range of numbers that can be a solution. In that range there is an infinite amount of possible numbers that make the inequality true. Part 1 Determine whether each number is a solution of the given inequality. 1) (a) (b) (c) (d) Part 2 Solve and graph the following inequalities. 2) 3)

6 4) 5) 6) 7)

7 Algebra I Pd Basic Inequalities HW 3D Solving and Graphing Inequalities (Day 2) Directions: All work is to be completed in your homework section of your spiral notebooks. 1) Determine whether each number is a solution of the given inequality: (a) -1 (b) 8 (c) 10 2) Determine whether each number is a solution of the given inequality: (a) 0 (b) -2 (c) -4 Directions: Solve and graph each inequality in your homework section of your notebooks. 3) 4) 5) 6) 7) 8) 9) 10) 11)

8 Algebra I Pd Basic Inequalities HW 3D HW ANSWERS 1) (a) no (b) yes (c) yes 2) (a) yes (b) yes (c) no 3) 4) 5) 6) 7) 8) 9) 10) 11)

9 Algebra I Pd Compound Inequalities 3E Graphing Compound Inequalities: Compound Inequalities Compound inequalities are problems that have more than one inequality that have to be graphed together. There are two different types we need to understand. AND ( ) problems and OR ( ) problems. AND: both must be true (for the number to be a part of the solution set, it must satisfy both parts of the compound inequality). Graph both and keep the INTERSECTION. Ex: OR: one must be true. (If the number satisfies either part of the compound inequality, or both parts, it is part of the solution set). Graph both and leave it alone. Ex: Part 1: Graph each of the compound inequalities. **(x < 2 and x < -3 can also be written as 3 < x < 2) 1) 2) 3) 4)

10 Part 2: Write a compound inequality that represents each phrase. Graph the solutions. 1) All real numbers that are less than or greater than or equal to. 2) The time a cake must bake is between 25 minutes and 30 minutes. Part 3: Solve and graph each of the compound inequalities. 1) 2) 3) 4)

11 Algebra I Pd Compound Inequalities HW 3F Compound Inequalities HW Part 1: Graph each of the compound inequalities in your spiral notebooks. 1) 2) Part 2: Write a compound inequality that represents each phrase. Graph the solutions. 3) All real numbers that are greater than and less than or equal to. 4) The time brownies must bake is between 15 minutes and 25 minutes. 5) All real numbers that are less than or greater than. Part 3: Solve and graph each of the compound inequalities. 6) 7) 8) 9) 10) 11)

12 Algebra I Pd Compound Inequalities HW 3F Homework Answers Part 1: Graph each of the compound inequalities in your spiral notebooks. 1) 2) Part 2: Write a compound inequality that represents each phrase. Graph the solutions. 3) All real numbers that are greater than and less than or equal to. 4) The time brownies must bake is between 15 minutes and 25 minutes. 5) All real numbers that are less than or greater than. Part 3: Solve and graph each of the compound inequalities. 6) 7) 8) 9) 10) 11)

13 Algebra I Pd Compound Inequalities (2) 3G Compound Inequalities Day 2 Part 1: Write a compound inequality for each of the graphs. 1) 2) 3) 4) Part 2: Solve and graph each of the compound inequalities. 5) 6)

14 7) 8) 9) 10) 11) 12)

15 Algebra I Pd Compound Inequalities (2) HW 3H Compound Inequalities Day 2 HW Part 1: Write a compound inequality for each of the graphs. 1) 2) 3) 4) Part 2: Solve and graph each of the compound inequalities. 5) 6) 7) 8) 9) 10)

16 Name Date Algebra I Pd Compound Inequalities (2) HW 3H Day 2 Homework Answers Part 1: Write a compound inequality for each of the graphs. 1) 2) 3) 4) Part 2: Solve and graph each of the compound inequalities. 5) 6) 7) 8) 9) 10)

17 Name Date Algebra I Pd Sets & Set Notation 3 I Sets - A set is any collection of objects, people or things defined as: {favorite day}, {favorite numbers} or {even numbers} - Each object in the set is called a member or an element of the set. Elements of a set never repeat. - Often the elements of a set are listed as a roster. - A roster is a list of the elements in a set, separated by commas and surrounded by curly braces {2, 4, 6, 8, 10} OR, elements can be given as a rule : {positive even integers less than 12} - EXAMPLES OF SET BUILDER NOTATION: {x: x > 7} "The set of all numbers, such that x is greater than 7" {x -5 < x < 1} "The set of numbers, such that x is greater than or equal to -5 and less than 1" - The symbol means is a member of or is an element of ex: 9 {odd numbers} The symbol means is not a member of ex: 7 {even numbers} Ex) Let set A be the numbers 3, 6, 9. A = {3, 6, 9} (in roster notation) Since 3 is in the set (an element) of A 3 A 5 is not in set (not an element) of A 5 A - Special Sets Finite set: the number of elements comes to an end {10, 11, , 99} Infinite set: the number of elements has no end {10, 11, 12...} Empty set: "null" or { } contains no numbers ex: the set of integers between 5 and 6 is { }. - If A = {0,1,2} then any set containing only elements of A is a subset of A ex: the following are all subsets of A {0} {1} {2} {0,1} {0,2} {1,2} {0,1,2} and note: {0,1,2} is called the improper subset itself & is a subset of all sets except for - Let B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {3, 6, 9} Set A is a subset of set B, since every element in set A is also an element of set B. The notation is: A B

18 Set Notation A set is a collection of unique elements. Elements in a set do not "repeat". By Interval Notation: An interval is a connected subset of numbers. Interval notation is an alternative to expressing your answer as an inequality. Unless specified otherwise, we will be working with real numbers. When using interval notation, the symbol: Means not included or open Means included or closed as an inequality. in interval notation. Inequality Graph Interval Notation Examples: 1) Which of the following is an infinite set? a) {x x is a United States Senator} b) {x x is an odd integer} c) {x x is a positive integer less than 1 million} d) {x x is a person born in 1982} 2) All numbers greater than or equal to positive ten." a) b) c) d) 3) "All numbers from negative three up to and including positive seven." a) b) c) d) 4) Which of the following is an empty set? a) {x x is a negative number greater than -20} b) {x x is an odd integer} c) {x x is a positive integer less than 1 million} d) {x x is an integer between 2 and 3} 5) Which statement below is the correct interval notation for line graph? a) b) c)

19 6) Which interval notation represents the set of all numbers from 2 through 7 inclusive? a) b) c) d) 7) Which statement below is the correct interval notation for line graph? a) b) c) d) 8) Which of the following represents the set of whole numbers? a) {x: x > 0} b) {x: x > 0} c) {x: x< 0} d) {x: x < 0} 9) What of the following represents the set of numbers that has the following two properties? i) The number must be at least 23. ii) The number must be less than 40. a) {x: 23 < x < 40} b) {x: 23 < x < 40} c) {x: 23 < x < 40} d) {x: 17 < x < 40} 10) Which statement below is the correct interval notation for line graph? a) b) c) d) 11) "All numbers from negative infinity up to and including four a) b) c) d) 12) What is the difference between the number line graphs of these two intervals: (2,8] and [2,8]? a) The first graph has an open circle at 2 while the second graph has a closed circle at 2. b) The first graph has a closed circle at 2 while the second graph has an open circle at 2. c) There is no difference between the graphs.

20 Algebra I Pd Sets & Set Notation HW 3J 1) Which of the following represents the set of positive real numbers? (a) {x: x 0} (b) {x: x > 0} (c) {x:x 0} (d) {x: x < 0} 2) Which of the following represents the set of non-positive real numbers? (a) {x: x 0} (b) {x: x > 0} (c) {x:x 0} (d) {x: x < 0} 3) Which of the following represents the set { -3, -2, -1, 0, 1, 2, 3 }? (a) {x: x is a real number} (b) {x: x is a rational number} (c) {x: x is an integer} 4) Which of the following represents the set {2,4,6, }? (a) {x: x = 2n, n= a positive integer} (b) {x: x = 2n, n = an even integer} (c) {x: x = n + 1, n = a positive integer} (d) {x: x = n + 1, n = an even integer} 5) Which of the following represents the set {x: x -3}? (a) (-3, ) (b) (-, -3) (c) [-3, ) (d) (-, -3] 6) Which of the following represents the solution to the inequality? (a) (, -2) (b) (-, -2] (c) (-, 2) (d) (-, 2] 7) Which of the following represents all the real numbers from 2 to 9, but not 2? (a) (2,9) (b) [2,9] (c) [2,9) (d) (2,9] 8) Which of the following represents the real numbers? (a) (-, ) (b) [-, ] (c) { } (d) 9) Monique states that the amount of money in her pocket is either greater than 17 dollars or at most 25 dollars. Which of the following represents the amount on money that is in Monique s pocket? (a) (17,25] (b) [0, ) (c) (17, ) (d) [0,25]

21 Algebra I Pd Sets Day 2 3K Sets The complement or is every element not in the set. If then The intersection of two sets is what elements they have in common if there are no elements in common the sets are said to be disjoint and the intersection is The union is all elements of both with no common elements listed twice. 1) Let U (the universal set) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {2, 4, 6, 8} B = {1, 2, 3, 4, 5} a) What is the complement of set A? b) What is the complement of set B? 2) From the set of positive integers less than 10, let set A = {prime numbers less than 10}. What is the complement of set A? 3) Let set A = and set B =. a) What is the set? b) What is the set?

22 4) Let A = {1, 2, 5, 6, 7} and C = {3, 4, 5}. What is the union of set A and set C. a) {1, 2, 5, 6, 7} b) {3, 4, 5} c) {1, 2, 3, 4, 5, 6, 7} d) {5} 5) Let A = {2, 4, 6, 7, 9} and C = {1, 3, 4, 7, 8}. What is the intersection of set A and set C. a) {2, 4, 6, 7, 9} b) {1, 3, 4, 7, 8} c) {4, 7} d) {1, 2, 3, 4, 6, 7, 8, 9} 6) Let A = {1, 2, 3, 4, 5, 6, 7} and C = {3, 4, 5}. What is the union of set A and set C. a) {1, 2, 3, 4, 5, 6, 7} b) {3, 4, 5} c) {1, 2, 6, 7} Part 2 Find each union or intersection. Let and 7) 8) 9) 10) 11) 12)

23 Algebra I Pd Sets Day 2 HW 3L 1) From the set of positive integers less than 20, let set A = {prime numbers less than 20}. What is the complement of set A? 2) From a standard deck of playing cards, let set N = {number cards}. How many cards are in set N c? 3) From the integers between -6 and 6, let set S = {multiples of 3}. What numbers are in the set S? 4) From a standard deck of playing cards, let set R consist of the following cards: R = {10 s, Jacks, Queens, Kings, Aces}. How many cards are in the set R? 5) From the set of Real Numbers, R, which of the following is the complement of the set of irrational numbers, I? (a) Rational Numbers (b) Natural Numbers (c) Integers (d) Counting Numbers 6) From the set {1,2,3,4,, 30}, let A = {Even numbers and not a multiple of 10}. If set B is a subset of set A where set B = {multiples of 3}, what numbers are in B c? 7) Let set A = {1,2,4,8,16} and let set B = {1,2,3,4,5,6,7,8,9}. What is 8) From a standard deck of playing cards, let set R = {all the red cards} and let set Q = {all the Queens}. How many cards are in the set? 9) Jessica wishes to buy a set of pens. She has two options. The first package of 4 pens contains the colors {Black, Red, Green, Blue} and the second package of 6 pens contains the colors {Blue, Purple, Green, Brown, Red, Yellow}. What color pens are in the intersection of the two packages? 10) Let set A = {-2,-1,0,1,2} and let set B = {0,1,2,3,4}. What is the set? 11) From a standard deck of playing cards, let B = {all the black cards} and let P = {all the picture cards}. How many cards are in set? (a) 6 (b) 28 (c) 32 (d) 42 12) Let set Q = {rational numbers} and let set I = {irrational numbers}. Which of the following represents the set? (a) The real numbers (b) The natural numbers (c) The integers (d) The null set 13) Let A = {1,3,5}, B = {-1,0,1}, and let C = {1,2,3}. Which of the following sets represents? (a) {1} (b) {-1,0,1,2,3,5} (c) {1,3} (d) {-1,1,3,5}

24 Algebra I Pd Sets Extra Practice 3G Directions: Place the final letter answer on the line provided to the left of each example. 1) All numbers greater than or equal to positive five." a) b) c) d) 2) "All numbers from negative six up to and including positive nine." a) b) c) d) 3) Which of the following is a finite set? a) {x x is an even integer} b) {x x is an odd integer} c) {x x is a integer less than 1 million} d) {x x is a student in this math class} 4) Which interval notation represents the set of all numbers from 2 through 7 inclusive? a) b) c) d) 5) Which of the following represents the set of natural numbers? a) {x: x > 0} b) {x: x > 0} c) {x: x< 0} d) {x: x < 0} 6) What of the following represents the set of numbers that has the following two properties? i) The number must be at least 19. ii) The number must be less than 60. a) {x: 19 < x < 60} b) {x: 19 < x < 60} c) {x: 19 < x < 60} d) {x: 19 < x < 60} 7) "All numbers from negative infinity up to and including two a) b) c) d) 8) Which of the following represents the set of negative real numbers? a){x: x 0} b) {x: x > 0} c) {x:x 0} d) {x: x < 0} 9) Which of the following represents the set {0, 1, 2, 3 }? a){x: x is a real number} b) {x: x is a whole number} c) {x: x is an integer} d) {x: x is a natural number} 10) Which of the following represents the set {3, 5, 7, }? a){x: x = 2n, n= a positive integer} b) {x: x = 2n, n = an even integer} c) {x: x = n + 1, n = a positive integer} d) {x: x = n + 1, n = an even integer} 11)Which of the following represents the set {x: x 5}? a) b) c) d) 12) Which of the following represents all the real numbers from 2 to 9, but not 2? a) b) c) d)

25 13) From the universal set of positive integers less than 15, let set A = {odd numbers less than 15}. What is the complement of set A? 14) From a standard deck of playing cards, let set N = {face cards}. How many cards are in set N c?

26 Algebra I Pd Inequalities and Sets Review Q2 Test 1 Topics: Solving Basic Inequalities Solving Complex Inequalities Sets Set Notation Directions: Solve and graph the following questions. Show all work on loose leaf. 1) 2) 3) 4) 5) 6) 7) 8) 9) or 10) 11) 12) 13) 14)

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