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Table of Contents

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

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  1. 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

  2. Key Terms Return to 

Table of 

Contents

  3. Key Terms Equation Inequality Solution Set Compound Inequality Set Builder Notation

  4. Pre-Assessment What do you already know? Return to 

Table of 

Contents

  5. 1 Evaluate the expression for a=2 and 
b=6. b - a

  6. 2 Compare and order. Write <, > or =. 1 [ ] 2 3 5   A < B > C =

  7. 3 Simplify the expression by combining like 
terms.  4x - 5 + 3x + 2 A 4x B 12x + 2 C 7x - 3 D 7x + 7

  8. 4 Simplify the expression (3 - d)5 A 3 - 5d B 15 - 5d C 15 + 5d D 15 - d

  9. 5 Solve r - 8 = -13

  10. 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

  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

  12. An inequality is a statement that two quantities 
are not equal. The quantities are compared by 
using one of the following signs:

  13. 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?

  14. 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?

  15. 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?

  16. Graphing and Writing Inequalities with One Variable Return to 

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  17. Graphing and Writing Inequalities with One Variable Objectives Identify solutions of inequalities 
with one variable. Write and graph inequalities with 
one variable.

  18. When you need to use an inequality to solve a 
word problem, you may encounter one of the 
phrases below.

  19. When you need to use an inequality to solve a word 
problem, you may encounter one of the phrases 
below.

  20. 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

  21. 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

  22. 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

  23. 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.

  24. 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

  25. 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.

  26. 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

  27. 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}

  28. 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". < >

  29. 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 ≥.

  30. Graphing Inequalities Do you know where to start? How do represent the starting point? Is there a special symbol?

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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 >.

  39. 8 -5 5 -4 -3 -2 -1 0 1 2 3 4 A x > 3 B x < 3 C x < 3 D x > 3

  40. 9 5 6 7 8 9 10 11 12 13 14 15 A 11 < x B 11 > x C 11 > x D 11 < x

  41. 10 -5 5 -4 -3 -2 -1 0 1 2 3 4 A x > -1 B x < -1 C x < -1 D x > -1

  42. 11 -5 5 -4 -3 -2 -1 0 1 2 3 4 A -4 < x B -4 > x C -4 < x D -4 > x

  43. 12 -5 5 -4 -3 -2 -1 0 1 2 3 4 A x > 0 B x < 0 C x < 0 D x > 0

  44. 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

  45. 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

  46. 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

  47. Simple Inequalities 
Involving Addition and Subtraction Return to 

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  48. Objectives Solve one-step inequalities by 
using addition. Solve one-step inequalities by 
using subtraction.

  49. 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!

  50. 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.

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