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

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

Answer:The solution is the set { all numbers greater than 77}. Example 1-1a SolveThen check your solution. Original inequality Add 12 to each side. This means all numbers greater than 77. CheckSubstitute 77, a number less than 77, and a number greater than 77.

Example 1-1b SolveThen check your solution. Answer:or { all numbers less than 14}

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

Example 1-2b SolveThen graph it on a number line. Answer:

Example 1-3a SolveThen graph the solution. Original inequality Subtract 23 from each side. Simplify. Answer:The solution set is

Example 1-3b SolveThen graph the solution. Answer:

Example 1-4a Then graph the solution. Original inequality Subtract 12n from each side. Simplify. Answer:Sinceis the same asthe solution set is

Example 1-4b Then graph the solution. Answer:

Example 1-5a 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 numberminustwo. 7n7n6n6n2>– Simplify. Subtract 6n from each side. Original inequality Answer:The solution set is

Example 1-5b 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 1-6a 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? WordsThe total cost of the two passes must be less than or equal to $100. VariableLetthe cost of the second pass. Inequality 100 The total cost is less than or equal to $100.

Example 1-6a Solve the inequality. Answer:The second pass must cost no more than $ Original inequality Subtract from each side. Simplify.

Example 1-6b 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

End of Lesson 1

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

Example 2-1a 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 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 2-1b Then check your solution. Answer:

Example 2-2a Original inequality Simplify. Multiply each side byand change Answer:The solution set is

Example 2-2b Answer:

Example 2-3a Write an inequality for the sentence below. Then solve the inequality. Four-fifths of a number is at most twenty. Four-fifthsofis at mosttwenty.a number r  20

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

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

Example 2-4a 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 2-4b Answer:

Example 2-5a using two methods. Method 1 Divide. Original inequality Divide each side by –8 and change. Simplify.

Example 2-5a Answer: The solution set is Method 2 Multiply by the multiplicative inverse. Original inequality Multiply each side by and change. Simplify.

Example 2-5b using two methods. Answer:

Example 2-6a Multiple-Choice Test Item Which inequality does not have the solution ABCD 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 A. D. C. B. Answer:B

Example 2-6b Multiple-Choice Test Item Which inequality does not have the solution? ABCD Answer:C

End of Lesson 2

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

Example 3-1a 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 Answer:Water will be a gas for all temperatures greater than 100°C. Original inequality Subtract 32 from each side. Simplify. Multiply each side by Simplify.

Example 3-1b Science The boiling point of helium is –452°F. Solve the inequalityto 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 3-2a Then check your solution. Original inequality Subtract 13 from each side. Simplify. Divide each side by –11 and change Simplify.

Example 3-2a Check To check the solution, substitute –6, a number less than –6, and a number greater than –6. Answer:The solution set is

Example 3-2b Then check your solution. Answer:

Example 3-3a 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 numberplus is less than a number minus three. twelve 4n4n+ < 12

Example 3-3a 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 3-3b 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 3-4a Original inequality Add c to each side. Simplify. Subtract 6 from each side. Simplify. Divide each side by 4. Simplify. Combine like terms. Distributive Property

Example 3-4a Answer: Sinceis the same as the solution set is

Example 3-4b Answer:

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

Example 3-5b Answer:Ø

End of Lesson 3

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

Example 4-1a Graph the solution set of Find the intersection. Graph

Example 4-1a Answer:The solution set isNote that the graph ofincludes the point 5. The graph ofdoes not include 12.

Example 4-1b Graph the solution set ofand

Example 4-2a 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. Graph Find the intersection.

Example 4-2a Answer: The solution set is

Example 4-2b Then graph the solution set. Answer:

Example 4-3a 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. WordsThe hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. VariablesLet 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 c or

Example 4-3a Now graph the solution set. Graph Find the union.

Example 4-3a Answer:

Example 4-3b 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 4-4a Then graph the solution set. or

Example 4-4a Graph Answer: Notice that the graph ofcontains every point in the graph ofSo, the union is the graph of The solution set is

Example 4-4b Then graph the solution set. Answer:

End of Lesson 4

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

Example 5-1a 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 Answer: The solution set is Writeasor Original inequality Subtract 6 from each side. Case 1Case 2 Simplify.

Example 5-1b Answer: {12, –2}

Example 5-2a 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 Answer:

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

Example 5-3a Then graph the solution set. Writeasand Original inequality Add 3 to each side. Simplify. Case 1 Case 2 Answer: The solution set is

Example 5-3b Then graph the solution set. Answer:

Example 5-4a Case 1 Case 2 Then graph the solution set. Writeasor Add 3 to each side. Simplify. Original inequality Divide each side by 3. Simplify.

Example 5-4a Answer: The solution set is

Example 5-4b Then graph the solution set. Answer:

End of Lesson 5

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

Example 6-1a 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 01 true 42 false 20 true 33 True or False yx

Example 6-1a 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 forin relation to the line.

Example 6-1b 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 6-2a 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 Step 2 Graph Sincedoes not include values when the boundary is not included in the solution set. The boundary should be drawn as a dashed line. Step 3 Select a point in one of the half-planes and test it. Let’s use (0, 0). Original inequality false y = 2x + 3

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

Example 6-2b Answer:

Example 6-3a 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 writeplustimes number of articles she can editis up to8 hours. hour x+8y

Example 6-3a 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 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.  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 6-3b 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 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.

End of Lesson 6

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