Notes for Algebra 1 Chapter 5

Solving Inequalities by Addition and Subtraction 5-1 Notes for Algebra 1 Solving Inequalities by Addition and Subtraction

Inequality An open sentence that contains <, >, < , or >

Addition Property of Inequality If the same number is added to both sides of a true inequality, the resulting inequality is also true. If 𝑎>𝑏, then 𝑎+𝑐>𝑏+𝑐 If 𝑎<𝑏, then 𝑎+𝑐<𝑏+𝑐

Example 1 pg. 285 Solve by adding Solve 𝑐−12>65. Check your answers

Example 1 pg. 285 Solve by adding Solve 𝑐−12>65. Check your answers 𝑐>77

Set-builder notation A more concise way of writing a solution set. Example 1 answer would be written as 𝑥 𝑥>77

Subtraction Property of Inequality If the same number is subtracted from each side of a true inequality, the resulting inequality is also true. If 𝑎>𝑏, then 𝑎−𝑐>𝑏−𝑐 If 𝑎<𝑏, then 𝑎−𝑐<𝑏−𝑐

Example 2 pg. 286 Solve by subtracting Solve the inequality. 𝑥+23<14

Example 2 pg. 286 Solve by subtracting Solve the inequality. 𝑥+23<14 𝑥 <−9

Example 3 pg. 287 Variables on each side. Solve 12𝑛−4 < 13𝑛 Graph the solution set

Example 3 pg. 287 Variables on each side. Solve 12𝑛−4 < 13𝑛 Graph the solution set 𝑛 𝑛 > −4 -5 -4 -3

Phrases for inequality < Less than, fewer than > Greater than, more than < at most, no more than, less than or equal to > at least, no less than, greater than or equal to.

Example 4 pg. 287 Use an inequality to solve a problem ENTERTAINMENT Tanya wants to buy season passes to two theme parks. If one season pass cost $54.99 and Tanya has $100 to spend on both passes, the second season pass must cost no more than what amount?

Example 4 pg. 287 Use an inequality to solve a problem ENTERTAINMENT Tanya wants to buy season passes to two theme parks. If one season pass cost $54.99 and Tanya has $100 to spend on both passes, the second season pass must cost no more than what amount? The second season pass must cost no more than $45.01.

5-1 pg. 288 13-43o, 57-81(x3)

Solving Inequalities by Multiplication and Division 5-2 Notes for Algebra 1 Solving Inequalities by Multiplication and Division

Multiplication Property of Inequality If both sides of an inequality that is true are multiplied by a positive number, the resulting inequality is also true. If 𝑎<𝑏 & 𝑐>0, then 𝑎𝑐<𝑏𝑐. If 𝑎>𝑏 & 𝑐>0, then 𝑎𝑐>𝑏𝑐. If both sides of an inequality that is true are multiplied by a negative number, the direction of the inequality is reversed to make the resulting inequality also true. If 𝑎<𝑏 & 𝑐<0, then 𝑎𝑐>𝑏𝑐. If 𝑎>𝑏 & 𝑐<0, then 𝑎𝑐<𝑏𝑐.

Example 1 pg. 293 write and solve an inequality HIKING Mateo walks at a rate of 3 4 mile per hour. He knows that it is at least 9 miles to Onyx Lake. How long will it take Mateo to get there? Write and solve an inequality to find the time.

Example 1 pg. 293 write and solve an inequality HIKING Mateo walks at a rate of 3 4 mile per hour. He knows that it is at least 9 miles to Onyx Lake. How long will it take Mateo to get there? Write and solve an inequality to find the time. 3 4 𝑡 > 9; 𝑡 𝑡 > 12 ; It will take Mateo at least 12 hours.

Example 2 pg. 293 Solve by multiplying

Example 2 pg. 293 Solve by multiplying

Division Property of Inequalities If both sides of an inequality that is true are divided by a positive number, the resulting inequality is also true. If 𝑎<𝑏 & 𝑐>0, then 𝑎 𝑐 < 𝑏 𝑐 . If 𝑎>𝑏 & 𝑐>0, then 𝑎 𝑐 > 𝑏 𝑐 . If both sides of an inequality that is true are divided by a negative number, the direction of the inequality is reversed to make the resulting inequality also true. If 𝑎<𝑏 & 𝑐<0, then 𝑎 𝑐 > 𝑏 𝑐 . If 𝑎>𝑏 & 𝑐<0, then 𝑎 𝑐 < 𝑏 𝑐 .

Example 3 pg. 294 Divide to solve the inequality Solve each inequality 1.) 12𝑘 > 60 2.) −8𝑞<136

Example 3 pg. 294 Divide to solve the inequality Solve each inequality 1.) 12𝑘 > 60 𝑘 𝑘 > 5 2.) −8𝑞<136 𝑞 𝑞≻17

5-2 pg. 295 11-39o, 48-66(x3)

Solving Multi-step inequalities 5-3 Notes for Algebra 1 Solving Multi-step inequalities

Example 1 pg. 298 Solve a Multi-Step inequality FAXES Adriana has a budget of $115 for taxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 25+0.08𝑝 < 115.

Example 1 pg. 298 Solve a Multi-Step inequality FAXES Adriana has a budget of $115 for taxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 25+0.08𝑝 < 115. She can send at most 1125 pages

Example 2 pg. 298 Inequality involving a Negative Coefficient Solve 13−11𝑑 > 79 Check your solution

Example 2 pg. 298 Inequality involving a Negative Coefficient Solve 13−11𝑑 > 79 Check your solution 𝑑 𝑑 < −6

Example 3 pg. 299 Write and Solve an inequality Define a variable, write an inequality, and solve the problem. Then check your solution. Four times a number plus twelve is less than the number minus three.

Example 3 pg. 299 Write and Solve an inequality Define a variable, write an inequality, and solve the problem. Then check your solution. Four times a number plus twelve is less than the number minus three. Let 𝑛= the number; 4𝑛+12<𝑛−3; 𝑛 𝑛<−5

Example 4 pg. 299 Distributive Property Solve 6𝑐+3 2−𝑐 > −2𝑐+1. Check your solution

Example 4 pg. 299 Distributive Property Solve 6𝑐+3 2−𝑐 > −2𝑐+1. Check your solution 𝑐 𝑐 > −1

Example 5 pg. 300 Empty Set and all Reals Solve each inequality. Check your solution. 1.) −7 𝑘+4 +11𝑘 > 8𝑘−2 2𝑘+1 2.) 2 4𝑟+3 < 22+8 𝑟−2

Example 5 pg. 300 Empty Set and all Reals Solve each inequality. Check your solution. 1.) −7 𝑘+4 +11𝑘 > 8𝑘−2 2𝑘+1 ∅ 2.) 2 4𝑟+3 < 22+8 𝑟−2 𝑟 𝑟 is a real number

5-3 pg. 301 13-53o, 60-81(x3)

Solving Compound Inequalities 5-4 Notes for Algebra 1 Solving Compound Inequalities

Compound Inequality Inequalities containing the words “and” or “or”

Intersection A graph of “and” inequalities. Part of the graphs where the two solutions overlap

Example 1 pg. 306 Solve and Graph Intersection Solve 7<𝑧+2 < 11. Then graph the solution set.

Example 1 pg. 306 Solve and Graph Intersection Solve 7<𝑧+2 < 11. Then graph the solution set. 𝑧 5<𝑧 < 9 4 5 7 9 10

Union Graph of “or” inequalities. “or” inequalities are true if at least one of the inequalities is true.

Example 2 pg. 307 Write and graph a compound inequality TRAVEL A ski resort has several types of hotel rooms and 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 a guest would pay per night at the resort.

Example 2 pg. 307 Write and graph a compound inequality TRAVEL A ski resort has several types of hotel rooms and 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 a guest would pay per night at the resort. 𝑛 𝑛 < 89 𝑜𝑟 𝑛 > 109 , where 𝑛 is the amount a guest pays per night. 79 89 99 109 119

Example 3 pg. 308 Solve and Graph a Union Solve 4𝑘−7 < 25 or 12−9𝑘 > 30. Then graph the solution set. 𝑘 𝑘 < 8 −1 2 5 8 11

5-4 pg. 508 7-27o, 28-36, 45-72(x3)

Inequalities Involving Absolute Value 5-5 Notes for Algebra 1 Inequalities Involving Absolute Value

Cases for solving Absolute values inequalities < Case 1: The expression inside the absolute value symbols is nonnegative Case 2: The expression inside the absolute value symbols is negative

Example 1 pg. 312 Solve Absolute Value Inequalities < Solve each inequality and graph the solution set. 1.) 𝑛−3 < 12 2.) 𝑥+6 <−8

Example 1 pg. 312 Solve Absolute Value Inequalities < Solve each inequality and graph the solution set. 1.) 𝑛−3 < 12 𝑛 −9 < 𝑛 <15 2.) 𝑥+6 <−8 ∅ −15 −9 3 −15 21

Example 2 pg. 313 Apply Absolute Value Inequalities RAINFALL The average annual rainfall in California for the last 100 years is 23 inches. However the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California?

Example 2 pg. 313 Apply Absolute Value Inequalities RAINFALL The average annual rainfall in California for the last 100 years is 23 inches. However the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California? 𝑥 13 < 𝑥 < 33

Cases for solving Absolute values inequalities > Case 1: The expression inside the absolute value symbols is nonnegative Case 2: The expression inside the absolute value symbols is negative

Example 3 pg. 313 Solve Absolute Value Inequalities > Solve each inequality then graph the solution set. 1.) 3𝑦−3 >9 2.) 2𝑥+7 > −11

Example 3 pg. 313 Solve Absolute Value Inequalities > Solve each inequality then graph the solution set. 1.) 3𝑦−3 >9 𝑦 𝑦<−2 𝑜𝑟 𝑦>4 −5 −2 1 4 7 2.) 2𝑥+7 > −11 𝑥 𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 −5 0 5

5-5 pg. 314 9-29o, 30, 31-35o, 48-66(x3)

Graphing Inequalities in Two variables 5-6 Notes for Algebra 1 Graphing Inequalities in Two variables

Boundary An equation defines a BOUNDARY, which divides the plane into two half-planes. When the boundary is included it is a closed half-plane. When the boundary is not included, it is an open half-plane.

Graphing Linear Inequalities 1.) Graph the boundary. Use a Solid Line for < or > . Use a dashed line for < or >. 2.) Use a test point to determine which half-plane should be shaded. 3.) Shade the half-plane that contains the solution.

Example 1 pg. 317 Graph an Inequality <or>

Example 1 & 2 pg. 317 Graph an Inequality 5 6 4 3 2 1 -4 -3 -2 -1

Example 3 pg. 318 Solve Inequalities from Graphs Use a graph to solve 2𝑥+3 < 7.

Example 3 pg. 318 Solve Inequalities from Graphs Use a graph to solve 2𝑥+3 < 7. 𝑥 < 2 5 4 3 2 1 -4 -3 -2 -1

Example 4 pg. 319 Write and Solve an Inequality Joe writes and edits short articles for a local newspaper. It takes him about an hour to write and article and about a half-hour to edit an article In Joe works up to 8 hours a day , how many articles can he write and edit in one day?

Example 4 pg. 319 Write and Solve an Inequality Joe writes and edits short articles for a local newspaper. It takes him about an hour to write and article and about a half-hour to edit an article In Joe works up to 8 hours a day , how many articles can he write and edit in one day? 𝑒 < −2𝑤+16

5-6 pg. 320 13-43, 44, 51-63(x3)