Splash Screen

Lesson 6-1 Solving Inequalities by Addition and Subtraction Lesson 6-2 Solving Inequalities by Multiplication and Division Lesson 6-3 Solving Multi-Step Inequalities Lesson 6-4 Solving Compound Inequalities Lesson 6-5 Solving Open Sentences Involving Absolute Value Lesson 6-6 Graphing Inequalities in Two Variables Contents

Example 1 Solve by Adding Example 2 Graph the Solution Example 3 Solve by Subtracting Example 4 Variables on Both Sides Example 5 Write and Solve an Inequality Example 6 Write an Inequality to Solve a Problem Lesson 1 Contents

Are inequalities used to describe school sports? Why Are inequalities used to describe school sports? Page 318 Lesson 1 Contents

Solve Then check your solution. Obj.: Solve linear inequalities by addition Solve Then check your solution. Original inequality Add 12 to each side. This means all numbers greater than 77. Check Substitute 77, a number less than 77, and a number greater than 77. Answer: The solution is the set {all numbers greater than 77}. Example 1-1a

Solve Then check your solution. Obj.: Solve linear inequalities by addition Solve Then check your solution. Answer: or {all numbers less than 14} Example 1-1b

Solve Then graph it on a number line. Obj.: Solve linear inequalities by addition Solve Then graph it on a number line. Original inequality Add 9 to each side. Simplify. Answer: Since is the same as y  21, the solution set is The heavy arrow pointing to the left shows that the inequality includes all the numbers less than 21. The dot at 21 shows that 21 is included in the inequality. Example 1-2a

Solve Then graph it on a number line. Obj.: Solve linear inequalities by addition Solve Then graph it on a number line. Answer: Example 1-2b

Solve Then graph the solution. Obj.: Solve linear inequalities by subtraction Solve Then graph the solution. Original inequality Subtract 23 from each side. Simplify. Answer: The solution set is Example 1-3a

Solve Then graph the solution. Obj.: Solve linear inequalities by subtraction Solve Then graph the solution. Answer: Example 1-3b

Then graph the solution. Obj.: Solve linear inequalities by subtraction Then graph the solution. Original inequality Subtract 12n from each side. Simplify. Answer: Since is the same as the solution set is Example 1-4a

Then graph the solution. Obj.: Solve linear inequalities by subtraction Then graph the solution. Answer: Example 1-4b

Obj.: Solve linear inequalities by subtraction Write an inequality for the sentence below. Then solve the inequality. Seven times a number is greater than 6 times that number minus two. Seven times a number is greater than six times that number minus two. 7n 6n 2 > – Original inequality Subtract 6n from each side. Simplify. Answer: The solution set is Example 1-5a

Write an inequality for the sentence below. Then solve the inequality. Obj.: Solve linear inequalities by subtraction Write an inequality for the sentence below. Then solve the inequality. Three times a number is less than two times that number plus 5. Answer: Example 1-5b

Obj.: Solve linear inequalities by subtraction Entertainment Alicia wants to buy season passes to two theme parks. If one season pass cost $54.99, and Alicia has $100 to spend on passes, the second season pass must cost no more than what amount? Words The total cost of the two passes must be less than or equal to $100. Variable Let the cost of the second pass. Inequality 100 The total cost is less than or equal to $100. Example 1-6a

Subtract 54.99 from each side. Obj.: Solve linear inequalities by subtraction Solve the inequality. Original inequality Subtract 54.99 from each side. Simplify. Answer: The second pass must cost no more than $45.01. Example 1-6a

Obj.: Solve linear inequalities by subtraction Michael scored 30 points in the four rounds of the free throw contest. Randy scored 11 points in the first round, 6 points in the second round, and 8 in the third round. How many points must he score in the final round to surpass Michael’s score? Answer: 6 points Example 1-6b

Are inequalities used to describe school sports? Why Are inequalities used to describe school sports? Page 318 Lesson 1 Contents

End of Lesson 1

Example 1 Multiply by a Positive Number Example 2 Multiply by a Negative Number Example 3 Write and Solve an Inequality Example 4 Divide by a Positive Number Example 5 Divide by a Negative Number Example 6 The Word “not” Lesson 2 Contents

Are inequalities important in landscaping? Why Are inequalities important in landscaping? Page 325 Lesson 1 Contents

Then check your solution. Obj.: Solve linear inequalities by multiplication Then check your solution. Original inequality Multiply each side by 12. Since we multiplied by a positive number, the inequality symbol stays the same. Simplify. Example 2-1a

Answer: The solution set is Obj.: Solve linear inequalities by multiplication Check To check this solution, substitute 36, a number less that 36 and a number greater than 36 into the inequality. Answer: The solution set is Example 2-1a

Then check your solution. Obj.: Solve linear inequalities by multiplication Then check your solution. Answer: Example 2-1b

Multiply each side by and change Obj.: Solve linear inequalities by multiplication Original inequality Multiply each side by and change Simplify. Answer: The solution set is Example 2-2a

Obj.: Solve linear inequalities by multiplication Answer: Example 2-2b

Obj.: Solve linear inequalities by multiplication Write an inequality for the sentence below. Then solve the inequality. Four-fifths of a number is at most twenty. Four-fifths of is at most twenty. a number r  20 Example 2-3a

Multiple each side by and do not change the inequality’s direction. Obj.: Solve linear inequalities by multiplication Original inequality Multiple each side by and do not change the inequality’s direction. Simplify. Answer: The solution set is . Example 2-3a

Write an inequality for the sentence below. Then solve the inequality. Obj.: Solve linear inequalities by multiplication Write an inequality for the sentence below. Then solve the inequality. Two-thirds of a number is less than 12. Answer: Example 2-3b

Answer: The solution set is Obj.: Solve linear inequalities by division Original inequality Divide each side by 12 and do not change the direction of the inequality sign. Simplify. Check Answer: The solution set is Example 2-4a

Obj.: Solve linear inequalities by division Answer: Example 2-4b

Divide each side by –8 and change < to >. Obj.: Solve linear inequalities by division using two methods. Method 1 Divide. Original inequality Divide each side by –8 and change < to >. Simplify. Example 2-5a

Method 2 Multiply by the multiplicative inverse. Obj.: Solve linear inequalities by multiplication Method 2 Multiply by the multiplicative inverse. Original inequality Multiply each side by and change < to >. Simplify. Answer: The solution set is Example 2-5a

using two methods. Answer: Obj.: Solve linear inequalities by division/multiplying using two methods. Answer: Example 2-5b

Multiple-Choice Test Item Which inequality does not have the solution Obj.: Solve linear inequalities by division/multiplying Multiple-Choice Test Item Which inequality does not have the solution A B C D Read the Test Item You want to find the inequality that does not have the solution set Solve the Test Item Consider each possible choice. Example 2-6a

Obj.: Solve linear inequalities by division/multiplying C. D. Answer: B Example 2-6a

Multiple-Choice Test Item Obj.: Solve linear inequalities by division/multiplying Multiple-Choice Test Item Which inequality does not have the solution ? A B C D Answer: C Example 2-6b

Are inequalities important in landscaping? Why Are inequalities important in landscaping? Page 325 Lesson 1 Contents

End of Lesson 2

Example 1 Solve a Real-World Problem Example 2 Inequality Involving a Negative Coefficient Example 3 Write and Solve an Inequality Example 4 Distributive Property Example 5 Empty Set Lesson 3 Contents

Are linear inequalities used in science? How Are linear inequalities used in science? Page 332 Lesson 1 Contents

Science The inequality F > 212 represents the Obj.: Solve linear inequalities with more than one operation. Science The inequality F > 212 represents the temperatures in degrees Fahrenheit for which water is a gas (steam). Similarly, the inequality represents the temperatures in degrees Celsius for which water is a gas. Find the temperature in degrees Celsius for which water is a gas. Example 3-1a

Subtract 32 from each side. Obj.: Solve linear inequalities with more than one operation. Original inequality Subtract 32 from each side. Simplify. Multiply each side by Simplify. Answer: Water will be a gas for all temperatures greater than 100°C. Example 3-1a

Science The boiling point of helium is –452°F. Solve Obj.: Solve linear inequalities with more than one operation. Science The boiling point of helium is –452°F. Solve the inequality to find the temperatures in degrees Celsius for which helium is a gas. Answer: Helium will be a gas for all temperatures greater than –268.9°C. Example 3-1b

Then check your solution. Obj.: Solve linear inequalities with more than one operation. Then check your solution. Original inequality Subtract 13 from each side. Simplify. Divide each side by –11 and change Simplify. Example 3-2a

Answer: The solution set is Obj.: Solve linear inequalities with more than one operation. Check To check the solution, substitute –6, a number less than –6, and a number greater than –6. Answer: The solution set is Example 3-2a

Then check your solution. Obj.: Solve linear inequalities with more than one operation. Then check your solution. Answer: Example 3-2b

Write an inequality for the sentence below. Then solve the inequality. Obj.: Solve linear inequalities with more than one operation. Write an inequality for the sentence below. Then solve the inequality. Four times a number plus twelve is less than a number minus three. Four times a number plus is less than a number minus three. twelve 4n + < 12 Example 3-3a

Subtract n from each side. Obj.: Solve linear inequalities with more than one operation. Original inequality Subtract n from each side. Simplify. Subtract 12 from each side. Simplify. Divide each side by 3. Simplify. Answer: The solution set is Example 3-3a

Write an inequality for the sentence below. Then solve the inequality. Obj.: Solve linear inequalities with more than one operation. Write an inequality for the sentence below. Then solve the inequality. 6 times a number is greater than 4 times the number minus 2. Answer: Example 3-3b

Distributive Property Combine like terms. Add c to each side. Obj.: Solve linear inequalities with the Distributive Property Original inequality Distributive Property Combine like terms. Add c to each side. Simplify. Subtract 6 from each side. Simplify. Divide each side by 4. Simplify. Example 3-4a

Answer: Since is the same as the solution set is Obj.: Solve linear inequalities with the Distributive Property Answer: Since is the same as the solution set is Example 3-4a

Answer: Obj.: Solve linear inequalities with the Distributive Property Example 3-4b

Distributive Property Obj.: Solve linear inequalities with the Distributive Property Original inequality Distributive Property Combine like terms. Subtract 4s from each side. This statement is false. Answer: Since the inequality results in a false statement, the solution set is the empty set Ø. Example 3-5a

Obj.: Solve linear inequalities with the Distributive Property Answer: Ø Example 3-5b

Are linear inequalities used in science? How Are linear inequalities used in science? Page 332 Lesson 1 Contents

End of Lesson 3

Example 1 Graph an Intersection Example 2 Solve and Graph an Intersection Example 3 Write and Graph a Compound Inequality Example 4 Solve and Graph a Union Lesson 4 Contents

Are compound inequalities used in tax tables? How Are compound inequalities used in tax tables? Page 339 Lesson 1 Contents

Graph the solution set of Obj.: Solve compound inequalities containing the word and and graph their solution sets. Graph the solution set of Graph Graph Find the intersection. Example 4-1a

Obj.: Solve compound inequalities containing the word and and graph their solution sets. Answer: The solution set is Note that the graph of includes the point 5. The graph of does not include 12. Example 4-1a

Graph the solution set of and Obj.: Solve compound inequalities containing the word and and graph their solution sets. Graph the solution set of and Example 4-1b

Then graph the solution set. Obj.: Solve compound inequalities containing the word and and graph their solution sets. Then graph the solution set. First express using and. Then solve each inequality. and Example 4-2a

The solution set is the intersection of the two graphs. Obj.: Solve compound inequalities containing the word and and graph their solution sets. The solution set is the intersection of the two graphs. Graph Graph Find the intersection. Example 4-2a

Answer: The solution set is Obj.: Solve compound inequalities containing the word and and graph their solution sets. Answer: The solution set is Example 4-2a

Then graph the solution set. Obj.: Solve compound inequalities containing the word and and graph their solution sets. Then graph the solution set. Answer: Example 4-2b

Variables Let c be the cost of staying at the resort per night. Obj.: Solve compound inequalities containing the word or and graph their solution sets. Travel A ski resort has several types of hotel rooms and several types of cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount that a quest would pay per night at the resort. Words The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Variables Let c be the cost of staying at the resort per night. Inequality Cost per night is at most $89 or the cost is at least $109. c 89 109 or Example 4-3a

Now graph the solution set. Obj.: Solve compound inequalities containing the word or and graph their solution sets. Now graph the solution set. Graph Graph Find the union. Example 4-3a

Obj.: Solve compound inequalities containing the word or and graph their solution sets. Answer: Example 4-3a

Answer: where c is the cost per seat Obj.: Solve compound inequalities containing the word or and graph their solution sets. Ticket Sales A professional hockey arena has seats available in the Lower Bowl level that cost at most $65 per seat. The arena also has seats available at the Club Level and above that cost at least $80 per seat. Write and graph a compound inequality that describes the amount a spectator would pay for a seat at the hockey game. Answer: where c is the cost per seat Example 4-3b

Then graph the solution set. Obj.: Solve compound inequalities containing the word or and graph their solution sets. Then graph the solution set. or Example 4-4a

Obj.: Solve compound inequalities containing the word or and graph their solution sets. Answer: Notice that the graph of contains every point in the graph of So, the union is the graph of The solution set is Example 4-4a

Then graph the solution set. Obj.: Solve compound inequalities containing the word or and graph their solution sets. Then graph the solution set. Answer: Example 4-4b

Are compound inequalities used in tax tables? How Are compound inequalities used in tax tables? Page 339 Lesson 1 Contents

End of Lesson 4

Example 1 Solve an Absolute Value Equation Example 2 Write an Absolute Value Equation Example 3 Solve an Absolute Value Inequality (<) Example 4 Solve an Absolute Value Inequality (>) Lesson 5 Contents

Is absolute value used in election polls? How Is absolute value used in election polls? Page 345 Lesson 1 Contents

The distance from –6 to –11 is 5 units. Obj.: Solve absolute value equations. Method 1 Graphing means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction. The distance from –6 to –11 is 5 units. The distance from –6 to –1 is 5 units. Answer: The solution set is Example 5-1a

Method 2 Compound Sentence Obj.: Solve absolute value equations. Method 2 Compound Sentence Write as or Case 1 Case 2 Original inequality Subtract 6 from each side. Simplify. Answer: The solution set is Example 5-1a

Obj.: Solve absolute value equations. Answer: {12, –2} Example 5-1b

Write an equation involving the absolute value for the graph. Obj.: Solve absolute value equations. Write an equation involving the absolute value for the graph. Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1. The distance from 1 to –4 is 5 units. The distance from 1 to 6 is 5 units. So, an equation is . Example 5-2a

Check Substitute –4 and 6 into Obj.: Solve absolute value equations. Answer: Check Substitute –4 and 6 into Example 5-2a

Write an equation involving the absolute value for the graph. Obj.: Solve absolute value equations. Write an equation involving the absolute value for the graph. Answer: Example 5-2b

Then graph the solution set. Obj.: Solve absolute value inequalities. Then graph the solution set. Write as and Case 1 Case 2 Original inequality Add 3 to each side. Simplify. Answer: The solution set is Example 5-3a

Then graph the solution set. Obj.: Solve absolute value inequalities. Then graph the solution set. Answer: Example 5-3b

Then graph the solution set. Obj.: Solve absolute value inequalities. Then graph the solution set. Write as or Case 1 Case 2 Original inequality Add 3 to each side. Simplify. Divide each side by 3. Simplify. Example 5-4a

Answer: The solution set is Obj.: Solve absolute value inequalities. Answer: The solution set is Example 5-4a

Then graph the solution set. Obj.: Solve absolute value inequalities. Then graph the solution set. Answer: Example 5-4b

Is absolute value used in election polls? How Is absolute value used in election polls? Page 345 Lesson 1 Contents

End of Lesson 5

Example 1 Ordered Pairs that Satisfy an Inequality Example 2 Graph an Inequality Example 3 Write and Solve an Inequality Lesson 6 Contents

Are inequalities used in budgets? How Are inequalities used in budgets? Page 352 Lesson 1 Contents

Obj.: Graph inequalities on the coordinate plane. From the set {(3, 3), (0, 2), (2, 4), (1, 0)}, which ordered pairs are part of the solution set for Use a table to substitute the x and y values of each ordered pair into the inequality. false 1 true 4 2 3 True or False y x Example 6-1a

Obj.: Graph inequalities on the coordinate plane. Answer: The ordered pairs {(3, 3), (2, 4)} are part of the solution set of . In the graph, notice the location of the two ordered pairs that are solutions for in relation to the line. Example 6-1a

Obj.: Graph inequalities on the coordinate plane. From the set {(0, 2), (1, 3), (4, 17), (2, 1)}, which ordered pairs are part of the solution set for Answer: {(1, 3), (2, 1)} Example 6-1b

Step 1 Solve for y in terms of x. Obj.: Graph inequalities on the coordinate plane. Step 1 Solve for y in terms of x. Original inequality Add 4x to each side. Simplify. Divide each side by 2. Simplify. Example 6-2a

Obj.: Graph inequalities on the coordinate plane. Step 2 Graph Since does not include values when the boundary is not included in the solution set. The boundary should be drawn as a dashed line. y = 2x + 3 Step 3 Select a point in one of the half-planes and test it. Let’s use (0, 0). Original inequality false Example 6-2a

Obj.: Graph inequalities on the coordinate plane. Answer: Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. y = 2x + 3 Example 6-2a

Check Test the point in the other half-plane, for example, (–3, 1). Obj.: Graph inequalities on the coordinate plane. Answer: Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. Check Test the point in the other half-plane, for example, (–3, 1). y = 2x + 3 Original inequality Since the statement is true, the half-plane containing (–3, 1) should be shaded. The graph of the solution is correct. Example 6-2a

Obj.: Graph inequalities on the coordinate plane. Answer: Example 6-2b

Obj.: Solve real-world problems involving linear inequalities. Journalism Lee Cooper writes and edits short articles for a local newspaper. It generally takes her an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a day, how many articles can she write and edit in one day? Step 1 Let x equal the number of articles Lee can write. Let y equal the number of articles that Lee can edit. Write an open sentence representing the situation. Number of articles she can write plus times number of articles she can edit is up to 8 hours. hour x + 8 y Example 6-3a

Step 2 Solve for y in terms of x. Obj.: Solve real-world problems involving linear inequalities. Step 2 Solve for y in terms of x. Original inequality Subtract x from each side. Simplify. Multiply each side by 2. Simplify. Example 6-3a

Obj.: Solve real-world problems involving linear inequalities. Step 3 Since the open sentence includes the equation, graph as a solid line. Test a point in one of the half-planes, for example, (0, 0). Shade the half-plane containing (0, 0) since is true. Answer: Example 6-3a

Step 4 Examine the situation. Obj.: Solve real-world problems involving linear inequalities. Step 4 Examine the situation.  Lee cannot work a negative number of hours. Therefore, the domain and range contain only nonnegative numbers.  Lee only wants to count articles that are completely written or completely edited. Thus, only points in the half-plane whose x- and y- coordinates are whole numbers are possible solutions.  One solution is (2, 3). This represents 2 written articles and 3 edited articles. Example 6-3a

Obj.: Solve real-world problems involving linear inequalities. Food You offer to go to the local deli and pick up sandwiches for lunch. You have $30 to spend. Chicken sandwiches cost $3.00 each and tuna sandwiches are $1.50 each. How many sandwiches can you purchase for $30? Answer: Example 6-3b

Obj.: Solve real-world problems involving linear inequalities. The open sentence that represents this situation is where x is the number of chicken sandwiches, and y is the number of tuna sandwiches. One solution is (4, 10). This means that you could purchase 4 chicken sandwiches and 10 tuna sandwiches. Example 6-3b

Are inequalities used in budgets? How Are inequalities used in budgets? Page 352 Lesson 1 Contents

End of Lesson 6