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Table of Contents. Key Terms. Pre-Assessment. Graphing/Writing with One Variable. Simple Inequalities Addition/Subtraction. Simple Inequalities Multiplication/Division. Two-Step & Multiple-Step. Compound Inequalities. Special Cases of Compound Inequalities. Key Terms.
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Table of Contents Key Terms Pre-Assessment Graphing/Writing with One Variable Simple Inequalities Addition/Subtraction Simple Inequalities Multiplication/Division Two-Step & Multiple-Step Compound Inequalities Special Cases of Compound Inequalities
Key Terms Return to Table of Contents
Key Terms Equation Inequality Solution Set Compound Inequality Set Builder Notation
Pre-Assessment What do you already know? Return to Table of Contents
1 Evaluate the expression for a=2 and b=6. b - a
2 Compare and order. Write <, > or =. 1 [ ] 2 3 5 A < B > C =
3 Simplify the expression by combining like terms. 4x - 5 + 3x + 2 A 4x B 12x + 2 C 7x - 3 D 7x + 7
4 Simplify the expression (3 - d)5 A 3 - 5d B 15 - 5d C 15 + 5d D 15 - d
5 Solve r - 8 = -13
6 There are 461 students and 20 teachers taking buses on a trip to a museum. Each bus can seat a maximum of 52. What is the least number of buses needed for the trip? A 8 B 9 C 10 D 11
What do these symbols mean? Less Than Greater Than Less Than or Equal To Greater Than or Equal To move square to reveal answer
An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:
When am I ever going to use it? Your parents and grandparents want you to start eating a healthy breakfast. The table shows the nutritional requirements for a healthy breakfast cereal with milk. Did you compare your favorite cereal's nutritional values to the healthy requirements on the table? Answer Healthy Breakfast Cereals (per serving) If you did, you found out that you have been eating a healthy breakfast. Now you can prove it to your parents and grandparents. 1. Suppose your favorite cereal has 2 grams of fat, 7 grams of protein, 3 grams of fiber and 4 grams of sugar. Is it a healthy cereal?
Healthy Breakfast Cereals (per serving) A cereal with 3 grams of fiber just makes it at being healthy. It needed at least 3 grams. Answer 2. Is a cereal with 3 grams of fiber considered healthy?
Healthy Breakfast Cereals (per serving) Answer A cereal with 5 grams of sugar is still considered healthy but is very close to being unhealthy. 3. Is a cereal with 5 grams of sugar considered healthy?
Graphing and Writing Inequalities with One Variable Return to Table of Contents
Graphing and Writing Inequalities with One Variable Objectives Identify solutions of inequalities with one variable. Write and graph inequalities with one variable.
When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.
When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.
How are these inequalities read? 2 + 2 > 3 Two plus two is greater than 3 2 + 2 > 3 Two plus two is greater than or equal to 3 2 + 2 ≥ 4 Two plus two is greater than or equal to 4 2 + 2 < 5 Two plus two is less than 5 2 + 2 ≤ 5 Two plus two is less than or equal to 5 2 + 2 ≤ 4 Two plus two is less than or equal to 4
Writing inequalities Let's translate each statement into an inequality. words x is less than 10 translate to inequality statement 10 x < 20 is greater than or equal to y 20 > y
You try a few: 1. 14 > a 2. b ≤ 8 3. 6 < 20f 4. t + 9 ≥ 36 5. 7 + w ≤ 10 6. 19 - p ≥ 2 7. n < 12 8. s < 50 9. p > 275 1. 14 is greater than a 2. b is less than or equal to 8 3. 6 is less than the product of f and 20 4. The sum of t and 9 is greater than or equal to 36 5. 7 more than w is less than or equal to 10 6. 19 decreased by p is greater than or equal to 2 7. Fewer than 12 items 8. No more than 50 students 9. At least 275 people attended the play Answers
Do you speak math? Try to change the following expressions from English into math. Answer 2x ≤ 6 Twice a number is at most six. Answer 2 + x ≥ 4 Two plus a number is at least four.
Three less than a number is less than three times that number. x - 3 < 3x Answer The sum of two consecutive numbers is at least thirteen. Answer x + (x + 1) ≥ 13 Three times a number plus one is at least ten. 3x + 1 > 10 Answer
Solution Sets A solution to an inequality is NOT a single number. It will have more than one value. 10 8 9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 This would be read as the solution set is all numbers greater than or equal to negative 5.
Let's name the numbers that are solutions of the given inequality. r > 10 Which of the following are solutions? {5, 10, 15, 20} 10 > 10 is not true So, not a solution 5 > 10 is not true So, not a solution 15 > 10 is true So, 15 is a solution 20 > 10 is true So, 20 is a solution Answer: {15, 20} are solutions of the inequality r > 10
Let's try another one. 30 ≥ 5d; {4,5,6,7,8} 30 ≥ 5d 30 ≥ 5(5) 30 ≥ 25 30 ≥ 5d 30 ≥ 5(4) 30 ≥ 20 30 ≥ 5d 30 ≥ 5(6) 30 ≥ 30 30 ≥ 5d 30 ≥ 5(7) 30 ≥ 35 30 ≥ 5d 30 ≥ 5(8) 30 ≥ 40 Answer: {4,5,6}
Graphing Inequalities with Greater/Less Than or Equal To An open circle on a number shows that the number is not part of the solution. It is used with "greater than" and "less than". The word equal is not included. < > A closed circle on a number shows that the number is part of the solution. It is used with "greater than or equal to" and "less than or equal to". < >
Remember! Open circle means that number is not included in the solution set and is used to represent < or >. Closed circle means the solution set includes that number and is used to represent ≤ or ≥.
Graphing Inequalities Do you know where to start? How do represent the starting point? Is there a special symbol?
Graphing Inequalities Step 1: Figure out what the inequality solution requires. For example, rewrite x is less than one as x < 1. Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, an open circle since it represents the starting point for the inequality solution but is not part of the solution. -5 5 -4 -3 -2 -1 1 2 0 3 4
x < 1 Step 3: Draw an arrow on the number line showing all possible solutions. For numbers greater than the variable, shade to the right of the boundary point. For numbers less than the variable, shade to the left of the boundary point. -5 5 -4 -3 -2 -1 1 2 0 3 4 Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality. -5 5 -4 -3 -2 -1 1 2 0 3 4
Graphing Inequalities Step 1: Figure out what the inequality solution requires. For example, rewrite x is greater than or equal to one as x > 1. Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, a closed circle since it represents the starting point for the inequality solution and is a part of the solution. 10 8 9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
You try Graph the inequality x > 5 10 8 9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Graph the inequality -3 > x 10 8 9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Try these. Graph the inequalities. 1. x > 4 -5 -4 -3 -2 -1 1 2 5 0 3 4 2. x < -5 -5 -4 -3 -2 -1 1 2 5 0 3 4
Try these. State the inequality shown. 1. -5 -4 -3 -2 -1 1 2 5 0 3 4 2. -5 -4 -3 -2 -1 1 2 5 0 3 4
7 Would this solution set be x > 4? True False 10 8 9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Remember! Closed circle means the solution set includes that number and is used to represent ≤ or ≥. Open circle means that number is not included in the solution set and is used to represent < or >.
8 -5 5 -4 -3 -2 -1 0 1 2 3 4 A x > 3 B x < 3 C x < 3 D x > 3
9 5 6 7 8 9 10 11 12 13 14 15 A 11 < x B 11 > x C 11 > x D 11 < x
10 -5 5 -4 -3 -2 -1 0 1 2 3 4 A x > -1 B x < -1 C x < -1 D x > -1
11 -5 5 -4 -3 -2 -1 0 1 2 3 4 A -4 < x B -4 > x C -4 < x D -4 > x
12 -5 5 -4 -3 -2 -1 0 1 2 3 4 A x > 0 B x < 0 C x < 0 D x > 0
A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions. Let e represent an employee's wages. An employee earns e at least > $7.50 7.5 7.5 0 1 2 3 4 5 6 7 8 9 10
Try this: The speed limit on a road is 55 miles per hour. Define a variable, write an inequality and graph the solution. Let s = speed 0 < s < 55 Answer -10 0 10 20 30 40 50 60
13 The sign shown below is posted in front of a roller coaster ride at the Wadsworth County Fairgrounds. If h represents the height of a rider in inches, what is a correct translation of the statement on this sign? All riders MUST be at least 48 inches tall. A h < 48 B h > 48 C h ≤ 48 D h ≥ 48
Simple Inequalities Involving Addition and Subtraction Return to Table of Contents
Objectives Solve one-step inequalities by using addition. Solve one-step inequalities by using subtraction.
Who remembers how to solve an algebraic equation? x + 3 = 13 - 3 - 3 x = 10 Use the inverse of addition Does 10 + 3 = 13 13 = 13 Be sure to check your answer!
Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations.