Algebra I. Simple Inequalities Involving Addition and Subtraction. Slide 1 / 182 Slide 2 / 182. Slide 4 / 182. Slide 3 / 182.

Size: px
Start display at page:

Download "Algebra I. Simple Inequalities Involving Addition and Subtraction. Slide 1 / 182 Slide 2 / 182. Slide 4 / 182. Slide 3 / 182."

Transcription

1 Slide 1 / 182 Slide 2 / 182 lgebra I Solving & Graphing Inequalities Slide 3 / 182 Slide 4 / 182 Table of ontents Simple Inequalities ddition/subtraction click on the topic to go to that section Simple Inequalities Multiplication/ivision Two-Step and Multiple-Step Inequalities Solving ompound Inequalities Special ases of ompound Inequalities Graphing Linear Inequalities in Slope-Intercept Form Solving Systems of Inequalitites Glossary & Standards Simple Inequalities Involving ddition and Subtraction Return to Table of ontents Slide 5 / 182 Inequality Slide 6 / 182 What do these symbols mean? (when read from LEFT to RIGHT) n Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities. Less Than click Less Than or Equal To Greater Than click Greater Than or Equal To

2 Slide 7 / 182 Slide 8 / 182 Inequality Write an inequality for the sentence below: Three times a number, n, is less than 210. lick The sum of a number, n, and fifteen is greater than or equal to nine. lick Slide 9 / 182 Graphing Inequalities Remember! Open circle means that number is not included in the solution set and is used to represent < or >. Slide 10 / 182 Solving Inequalities 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. losed circle means the solution set includes that number and is used to represent or. Slide 11 / 182 Slide 12 / 182 Isolate the Variable To find the solution, isolate the variable x. Remember, it is isolated when it appears by itself on one side of the equation.

3 Slide 13 / 182 Slide 14 / 182 Solving Inequalities Step 2: ecide whether or not the circle on your boundary should be open or closed based on the symbol used. You can check the computation by substituting the end point of 6 for x. In this case, the end point is not included (open circle) since x < Slide 15 / 182 Review of Solving Inequalities Using ddition and Subtraction The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: Slide 16 / Which graph is the solution to the inequality: a number, n, minus is greater than one third? equations-inequalities-7th-grade/ Slide 17 / 182 Slide 18 / Which graph is the solution to the inequality? 3 Which graph is the solution to the inequality?

4 Slide 19 / Which graph is the solution to the inequality? Slide 20 / Which graph is the solution to the inequality? Slide 21 / 182 Slide 22 / 182 Inequalities Involving Multiplication and ivision Simple Inequalities Involving Multiplication and ivision gain, similarly to solving equations, we can use the properties of multiplication and division to solve and graph inequalities - with one minor difference, which we will encounter in the upcoming slides. Return to Table of ontents Slide 23 / 182 Slide 24 / 182 Multiplying or ividing by a Positive Number Since x is multiplied by 3, divide both sides by 3 to isolate the variable

5 Slide 25 / 182 Review of Solving Inequalities Using Multiplication and ivision The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: Slide 26 / Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24? equations-inequalities-7th-grade/ Slide 27 / 182 Slide 28 / 182 Slide 29 / 182 Slide 30 / Find the solution to the inequality. 10 Find the solution to the inequality.

6 Slide 31 / 182 Multiplying or ividing by a Negative Number Slide 32 / 182 Solve and Graph So far, all the operations we have used worked the same as solving equations. The difference between solving equations versus inequalities is revealed when multiplying or dividing by a negative number. *Note: ividing each side by changes the to. The direction of the inequality changes only if the number you are using to multiply or divide by is negative click for answer Slide 33 / 182 Slide 34 / Solve the inequality and graph the solution. 12 Solve the inequality and graph the solution Slide 35 / 182 Slide 36 / Solve the inequality and graph the solution. 14 Solve the inequality and graph the solution

7 Slide 37 / 182 Slide 38 / 182 Summary In review, an inequality symbol stays the same direction when you: dd, subtract, multiply or divide by the same positive number on both sides. dd or subtract the same negative number on both sides. n inequality symbol changes direction when you: Multiply or divide by the same negative number on both sides. Solving Two-Step and Multiple-Step Inequalities Return to Table of ontents Slide 39 / 182 Slide 40 / 182 Inequalities Now we'll solve more complicated inequalities that have multi-step solutions because they involve more than one operation. Solving inequalities is like solving a puzzle. Keep working through the steps until you get the variable you're looking for alone on one side of the inequality using the same strategies as solving an equation. Slide 41 / 182 Multiplying or ividing by a Negative Number Slide 42 / 182 nother reminder! If you multiply or divide by a negative number, reverse the direction of the inequality symbol!

8 Slide 43 / 182 Two Step Inequalities Example: Solve the inequality and graph the solution. Slide 44 / 182 Solve and Graph Try these. Solve each inequality and graph each solution. 1. dd 9 to both sides ivide both sides by 4 (sign stays the same) click for answer Slide 45 / 182 Slide 46 / 182 Solve and Graph 15 Solve and graph the solution. Try these. Solve each inequality and graph the solution Slide 47 / 182 Slide 48 / 182

9 Slide 49 / 182 Slide 50 / Solve and graph the solution Slide 51 / Which graph represents the solution set for: Question from P lgebra I End-of-ourse Practice Test 21 Slide 52 / 182 Find all negative odd integers that satisfy the following inequality. Select all that apply E F G H From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 53 / Which value of x is in the solution set of? Slide 54 / What is the solution of? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, 2011.

10 Slide 55 / 182 Slide 56 / In the set of positive integers, what is the solution set of the inequality? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 57 / Given: etermine all elements of set that are in the solution of the inequality Slide 58 / 182 Inequalities in the Real World Inequalities are helpful when applied to real life scenarios. These inequalities can be used for budgeting purposes, speed limits, cell phone data usage, and building materials management, just to name a few. Translating between the languages of English words to numbers/ symbols is imperative in being able to solve the correct inequality. The next slides will provide ample practice in setting up and solving these inequality applications. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 59 / 182 Slide 60 / 182 Write an Inequality and Solve Example #2: You have $65.00 in birthday money and want to buy some s and a V. Suppose a V cost $15.00 and a cost $ Write an inequality and solve to find out how many s you can buy along with one V.

11 Slide 61 / 182 Write an Inequality and Solve Example #3: Matt was getting ready to go back to school. He had $150 to buy school supplies. Matt bought 3 pairs of pants and spent $30 on snacks and other items. Slide 62 / 182 Write an Inequality and Solve Example #4: You have $60 to spend on a concert. Tickets cost $18 each and parking is $8. Write an inequality to model the situation. How many tickets can you buy? How much could one pair of pants cost, if they were all the same price? Write an inequality and solve. Slide 63 / 182 Write an Inequality and Solve Example #5: If you borrow the $60 from your mom and pay her back at a rate of $7 per week, when will your debt be under $15? Slide 64 / 182 Write an Inequality and Solve Example #6: To earn an in math class, you must earn a total of at least 180 points on three tests. On the first two tests, your scores were 58 and 59. What is the minimum score you must get on the third test in order to earn an? efine a variable, write an inequality and graph the solutions Slide 65 / 182 Slide 66 / 182 Write an Inequality and Solve Example #7: Thelma and Laura start a lawn-mowing business and buy a lawnmower for $225. They plan to charge $15 to mow one lawn. What is the minimum number of lawns they need to mow if they wish to earn a profit of at least $750? 27 Roger is having a picnic for 78 guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which inequality could be used to determine how many packages of hot dogs Roger will need to buy? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, 2011 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, 2011.

12 Slide 67 / school group needs a banner to carry in a parade. The narrowest street the parade is marching down measures 36 ft across, but some space is taken up by parked cars. The students have decided the banner should be 18 ft long. There is 45 ft of trim available to sew around the border of the banner. What is the greatest possible width for the banner? Slide 68 / dmission to a town fair is $7.00. You plan to spend $6.00 for lunch and $4.50 for snacks. Each ride costs $2.25. If you have $35 to spend, what is the number of rides you can go on? 6 rides 7 rides 8 rides 9 rides Slide 69 / female gymnast is participating in a 4-event competition. Each event is scored on a ten-point scale. She scored a 9.1 in uneven bars, an 8.5 on the balance beam, and a 9.4 on the vault. Which inequality represents the remaining score required in the floor exercise for the gymnast to receive at least an 8.9 average? r r 8.6 Slide 70 / 182 Solving ompound Inequalities r r 8.6 Return to Table of ontents Slide 71 / 182 Slide 72 / 182 ompound Inequalities When two inequalities are combined into one statement by the words N/OR, the result is called a compound inequality. solution of a compound inequality joined by and is any number that makes both inequalities true. solution of a compound inequality joined by or is any number that makes either inequality true.

13 Slide 73 / 182 Slide 74 / Which inequality is represented in the graph below? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 75 / Which inequality is represented in the graph below? Slide 76 / 182 Solving ompound Inequalities that contain an N statement N is the same as writing You will need to solve both of these inequalities and graph their intersection. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 77 / 182 Slide 78 / 182

14 Slide 79 / 182 Slide 80 / Which result below is correct for this inequality: 34 Which result below is correct for this inequality: 2 1 / / Slide 81 / 182 Slide 82 / Which result below is correct for this inequality: 36 Which result below is correct for this inequality: Slide 83 / 182 Slide 84 / Which result below is correct for this inequality:

15 Slide 85 / 182 Slide 86 / 182 Writing a ompound Inequality From a Graph How would you write this? Slide 87 / 182 Writing a ompound Inequality From a Graph Slide 88 / 182 ompound Inequalities Solve and graph the solution set How would you write this? 2. or Slide 89 / 182 Slide 90 / or ompound Inequalities Solve and graph the solution set. 38 In order to be admitted for a certain ride at an amusement park, a child must be greater than or equal to 36 inches tall and less than 48 inches tall. Which graph represents these conditions? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, 2011.

16 Slide 91 / 182 Slide 92 / Which graph represents the solution set for and? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 93 / 182 Slide 94 / Solve Slide 95 / 182 Slide 96 / 182

17 Slide 97 / 182 Slide 98 / 182 pplication of ompound Inequalities Let's start off by translating the words of an applied problem into math. The sum of 3 times a number and two lies between 8 and 11. "The sum of 3 times a number and two" translates into what? Slide 99 / 182 pplication of ompound Inequalities The sum of 3 times a number and two lies between 8 and 11. How will we translate "lies between 8 and 11"? Slide 100 / 182 pplication of ompound Inequalities cell phone plan offers free minutes for no more than 250 minutes per month. efine a variable and write an inequality for the possible number of free minutes. Graph the solution. What inequality symbol will we use? Why? What is the inequality? Solve and graph the inequality. Slide 101 / 182 Slide 102 / Each type of marine mammal thrives in a specific range of temperatures. The optimal temperatures for dolphins range from 50 F to 90 F. Which inequality represents the temperatures where dolphins will not thrive?

18 Slide 103 / store is offering a $50 mail in rebate on all color printers. Nathan is looking at different color printers that range in price from $165 to $275. How much can he expect to spend after the rebate? $115 x $225 x < $115 or x > $225 $215 x $325 x < $215 or x > $325 Slide 104 / One quarter of a number decreased by 7 is at most 11 or greater than 15. Which compound inequality represents the possible values of the number? Slide 105 / Lyla has scores of 82, 92, 93, and 99 on her math tests. Use a compound inequality to find the range of scores she can make on her final exam to receive a in the course. The final exam counts as two test grades, and a is received if the final course average is from 85 to 92. Slide 106 / 182 Special ases of ompound Inequalities Return to Table of ontents Slide 107 / 182 Slide 108 / 182 Special ases solution of a compound inequality joined by and is any number that makes both inequalities true. When there is no number that makes both inequalities true, we say there is no solution. When all numbers make both inequalities true, we say the solution is the set of Reals or ll Reals.

19 Slide 109 / 182 Slide 110 / 182 Special ases Solve each set of compound inequalities. 1. and 2. or Slide 111 / 182 Slide 112 / 182 Special ases Solve each set of compound inequalities. 3. and Graphing Linear Inequalities in Slope-Intercept Form 4. and Return to Table of ontents Slide 113 / 182 Slide 114 / 182 Graphing The following are graphs of linear inequalities. Shading is above the dotted line.this means the solutions are above the line but NOT on it. Shading is below the dotted line.this means the solutions are below the line but NOT on it.

20 Slide 115 / 182 Graphing The following are graphs of linear inequalities. Slide 116 / 182 How to Graph a Linear Inequality Shading is above a solid line.this means the solutions are above the line N on it. Shading is below a solid line. This means the solutions are below the line N on it. 1) ecide where the boundary goes: Solve inequality for y, for example y > 2x - 1 2) ecide whether boundary should be: - solid ( or : points on the boundary make the inequality true) or - dashed (< or >: points on the boundary make the inequality false) 3) Graph the boundary (the line). 4) ecide where to shade: y > or y : shade above (referring to y-axis) the boundary y < or y : shade below (referring to y-axis) the boundary Or, you can test a point Graph Slide 117 / 182 Graphing Step 1: Solve for y: (Think ), m = and b = 1 Step 2: The line should be dashed because the inequality is < Graph Step 1: Solve for y Slide 118 / 182 Graphing Step 3: Graph boundary Step 4: Shade below the boundary line because y < Step 2: The line should be solid because the inequality is Step 3: Graph boundary Step 4: Shade above the boundary line because y Graph Slide 119 / 182 Graphing Is the equation already solved for y? Is the line solid or dashed? Explain why this is the case. The line is dashed because it is not included in the inequality. click to reveal Slide 120 / Why are there dashed boundaries on some graphs of inequalities? Points on the line make the inequality false. Points on the line make the inequality true. The slope of the line depends on the line type. The y-intercept depends on the line type. Will we shade above or below the line? Explain why this is the case. You shade above the line because the inequality shows that y is greater than the expression on the right hand side. Or, if you test a point (0, 0), it satisfies the inequality, so click to reveal you shade in that direction. click to reveal the inequality graph

21 Slide 121 / For which of these inequalities would the graph have a solid boundary and be shaded above? Slide 122 / For which of these inequalities would the graph have a dashed boundary and be shaded above? Slide 123 / 182 Slide 124 / Which inequality is graphed? Slide 125 / Graph the solution set of. When you finish, type the number "1" into your responder. Slide 126 / 182 Modeling with Inequalities Throughout this unit, you have learned how to solve and graph inequalities, both on a number line and in the coordinate plane. We can apply these skills to solve realistic word problems, such as purchasing items at a store within a budget and earning money through various jobs. Let's get started. PR - EOY - Question #2 Non-alculator Section - SMRT Response Format

22 Slide 127 / 182 Modeling with Inequalities t a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants. Part Write an inequality that would be used to model the situation. Slide 128 / 182 Modeling with Inequalities t a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants. Part Write an inequality that would be used to model the situation. Part Graph the inequality in a coordinate plane. Part List 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget. Slide 129 / 182 Modeling with Inequalities t a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 y each. You have $125 to spend. Let x represents 20 the dress shirts and y represents the number of pairs of dress pants. 15 Part Graph the inequality in a coordinate plane. 10 Slide 130 / 182 Modeling with Inequalities t a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants. Part List 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget x Slide 131 / t a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs. Part Which inequality would be used to model this situation? Slide 132 / t a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs. Part Graph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder y x

23 Slide 133 / t a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs. Part Which pairs (x, y) can represent the amount of soccer balls and footballs purchased at the sports shop? Select all that apply. (7, 1) (2, 3) (4, 6) (3, 3) E (1, 4) Slide 134 / group of friends went to the movies on Friday night. fter purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the number of sodas purchased and y represent the buckets of popcorn purchased. Part Which inequality would be used to model this situation? Slide 135 / group of friends went to the movies on Friday night. fter purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the y number of sodas purchased and y 20 represent the buckets of popcorn purchased. 15 Part Graph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder x Slide 136 / group of friends went to the movies on Friday night. fter purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the number of sodas purchased and y represent the buckets of popcorn purchased. Part Which pairs (x, y) can represent the amount spent on soda and buckets of popcorn at the theater? Select all that apply. (17, 1) (10, 5) (8, 4) (5, 5) E (3, 7) Slide 137 / 182 Slide 138 / 182 Vocabulary Solving Systems of Inequalities system of linear inequalities is two or more linear inequalities. The solution to a system of linear inequalities is the intersection of the half-planes formed by each linear inequality. The most direct way to find the solution to a system of linear inequalities is to graph the equations on the same coordinate plane and find the region of intersection. Return to Table of ontents

24 Slide 139 / 182 Graphing a System of Linear Inequalities Slide 140 / 182 Example Solve the following system of linear inequalities. Step 1: Graph the boundary lines of each inequality. Remember: - dashed line for < and > - solid line for and Step 2: Shade the half-plane for each inequality. Step 1: 10 5 y Step 3: Identify the intersection of the half-planes. This is the solution to the system of linear inequalities x Slide 141 / 182 Slide 142 / 182 Example ontinued Example ontinued Step 2: y Step 3: y x x Slide 143 / 182 Example Solve the following system of linear inequalities. Slide 144 / 182 Example ontinued Step 1: 10 y Step 2: 10 y x x

25 Slide 145 / 182 Example ontinued Slide 146 / 182 Example Solve the following system of linear inequalities. Step 3: 10 y Step 1: 10 y x x Slide 147 / 182 Slide 148 / 182 Example ontinued Example ontinued Step 2: y Step 3: y x x Slide 149 / 182 Slide 150 / 182

26 Slide 151 / 182 Slide 152 / hoose the graph below that displays the solution to the following system of linear inequalities: Slide 153 / 182 Slide 154 / hoose the graph below that displays the solution to the following system of linear inequalities: Slide 155 / hoose the graph below that displays the solution to the following system of linear inequalities: Slide 156 / hoose all of the linear inequalities that correspond to the following graph:

27 Slide 157 / Which point is in the solution set of the system of inequalities shown in the accompanying graph? Slide 158 / Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? (0, 4) (, 1) (0, 0) (1, 5) (2, 4) (4, ) (0, 1) (3, 2) From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 159 / Which ordered pair is in the solution set of the following system of linear inequalities? Slide 160 / Mr. raun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. raun can buy? (0, 3) (2, 0) ( 1, 0) ( 1, 4) From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Slide 161 / system of inequalities is given. Slide 162 / 182 Modeling with a System of Inequalities Graph the solution set of the system of linear inequalities in the coordinate plane. When you finish, type the number "1" into your Responder. Similar to solving application problems by graphing a single inequality, we can also apply our skills with solving a system of inequalities to solve realistic word problems. Let's get started. PR - P - Question #3 Non-alculator Section - SMRT Response Format

28 Slide 163 / 182 Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part : Graph the solution set of the system of linear inequalities in a coordinate plane. Part : reate 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Part : Given the conditions in Part, if Preston mows lawns for 9 hours this month, what is the minimum number of hours he would have to work at the deli to earn at least $150? Give your answer to the nearest whole hour. Part : Given the conditions in Part, Preston prefers mowing lawns over working at the deli. What is the maximum number of hours he can mow lawns to be able to earn at least $150? Give your answer to the nearest whole hour. Slide 165 / 182 Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part : reate 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Slide 164 / 182 Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of y 15 hours per month. Let x represent the number 20 of hours Preston mows lawns and y represent the number of hours 15 Preston works at the deli. Part : Graph the solution set of the system of linear inequalities in a coordinate plane Slide 166 / Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part : Given the conditions in Part, if Preston mows lawns for 5 hours this month, what is the minimum number of hours he would have to work at the deli to earn at least $150? Give your answer to the nearest whole hour. x Slide 167 / 182 Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part : Given the conditions in Part, Preston prefers mowing lawns over working at the deli. What is the maximum number of hours he can mow lawns to be able to earn at least $150? Give your answer to the nearest whole hour. Slide 168 / Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin y knows that the will sell more than comic books. Let x represent the number of comic books sold 30 and y represent the packs of baseball cards sold. Part : Graph the solution set of the system of linear inequalities in a coordinate plane. When you finish, type the number "1" into your Responder x

29 Slide 169 / Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold. Part Which pairs (x, y) represent the sales of comic books and packs of baseball cards to meet the given conditions? Select all that apply. (25, 25) (26, 8) (30, 10) (35, 25) E (18, 40) Slide 170 / Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold. Part Given the conditions in Part, if Gavin sold 14 packs of baseball cards, what is the minimum number of comic books he would need to sell to earn at least $210? Give your answer to the nearest whole number. Slide 171 / Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part Graph the solution set of the system of linear inequalities in the coordinate plane. When you finish, type the number "1" into your Responder. PR - EOY - Question #25 alculator Section - SMRT Response Format Slide 173 / Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part Given the conditions in Part, if Leah babysits for 7 hours this month, what is the minimum number of hours she would have to work at the ice cream shop to earn at least $120? Give your answer to the nearest whole hour. Slide 172 / Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part Which pairs (x, y) represent hours that Leah could work to meet the given conditions? Select all that apply. (4, 15) (5, 12) (10, 9) (15, 5) E (19, 1) PR - EOY - Question #25 alculator Section Slide 174 / Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part Given the conditions in Part, Leah prefers babysitting over working at the ice cream store. What is the maximum number of hours she can babysit to be able to earn at least $120? Give your answer to the nearest whole hour. PR - EOY - Question #25 alculator Section PR - EOY - Question #25 alculator Section

30 Slide 175 / 182 Slide 176 / 182 Inequality Glossary & Standards Slide 177 / 182 Solution Set Return to Table of ontents ny number that, when substituted into an equation/inequality, will satisfy the equation/ inequality n Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities. 2 < 18 x > 6 x Slide 178 / 182 r ompound Inequality r 11 Two inequalities that are combined into one statement by the words N/OR ack to Instruction r - 9 = r = 11 {11} check: 11-9 = 2 2 = 2 r r 11 Solution is not included! Solution is included! x > N x < 3 < x < 3 "and" means intersection "or" means union x OR x 3 ack to Instruction ack to Instruction Slide 179 / 182 No Solution When there is no number that makes the equation/inequalities true Slide 180 / 182 Reals When all (any) numbers make the equation/inequalities true 2x + 8 = 2(x - 4) 2x + 8 = 2x = -8 2x 18 N x > 12 x 9 N x < { } or { } "no solution" 2x + 8 = 2(x + 4) 2x + 8 = 2x = 0 R x + 3 > 17 OR 5(x + 2) > 0 x -7 OR x > R "reals" "all real numbers" R ack to Instruction ack to Instruction

31 Slide 181 / 182 System of Linear Inequalities Slide 182 / 182 Throughout this unit, the Standards for Mathematical Practice are used. Two or more linear inequalities y > 2x - 3 y < -x + 4 y x ack to Instruction MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: onstruct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: ttend to precision. MP7: Look for & make use of structure. MP8: Look for and express regularity in repeated reasoning. dditional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

Algebra I. Slide 1 / 182. Slide 2 / 182. Slide 3 / 182. Solving & Graphing Inequalities. Table of Contents

Algebra I. Slide 1 / 182. Slide 2 / 182. Slide 3 / 182. Solving & Graphing Inequalities. Table of Contents Slide 1 / 182 Slide 2 / 182 lgebra I Solving & Graphing Inequalities 2016-01-11 www.njctl.org Table of ontents Slide 3 / 182 Simple Inequalities ddition/subtraction click on the topic to go to that section

More information

Simple Inequalities Involving Addition and Subtraction. Unit 3 Inequalities.notebook. November 18, Table of Contents

Simple Inequalities Involving Addition and Subtraction. Unit 3 Inequalities.notebook. November 18, Table of Contents Table of Contents Simple Inequalities Addition/Subtraction Simple Inequalities Multiplication/Division Two-Step and Multiple-Step Inequalities Solving Compound Inequalities Special Cases of Compound Inequalities

More information

Algebra I Solving & Graphing Inequalities

Algebra I Solving & Graphing Inequalities Slide 1 / 182 Slide 2 / 182 Algebra I Solving & Graphing Inequalities 2016-01-11 www.njctl.org Slide 3 / 182 Table of Contents Simple Inequalities Addition/Subtraction click on the topic to go to that

More information

Algebra I. Systems of Linear Equations and Inequalities. 8th Grade Review. Slide 1 / 179 Slide 2 / 179. Slide 4 / 179. Slide 3 / 179.

Algebra I. Systems of Linear Equations and Inequalities. 8th Grade Review. Slide 1 / 179 Slide 2 / 179. Slide 4 / 179. Slide 3 / 179. Slide 1 / 179 Slide 2 / 179 lgebra I Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Slide 3 / 179 Table of Contents Click on the topic to go to that section 8th Grade Review of Systems

More information

UNIT 5 INEQUALITIES CCM6+/7+ Name: Math Teacher:

UNIT 5 INEQUALITIES CCM6+/7+ Name: Math Teacher: UNIT 5 INEQUALITIES 2015-2016 CCM6+/7+ Name: Math Teacher: Topic(s) Page(s) Unit 5 Vocabulary 2 Writing and Graphing Inequalities 3 8 Solving One-Step Inequalities 9 15 Solving Multi-Step Inequalities

More information

8th Grade. Radical Expressions Containing Variables. Slide 1 / 87 Slide 2 / 87. Slide 3 / 87. Slide 4 / 87. Slide 5 / 87. Slide 5 (Answer) / 87

8th Grade. Radical Expressions Containing Variables. Slide 1 / 87 Slide 2 / 87. Slide 3 / 87. Slide 4 / 87. Slide 5 / 87. Slide 5 (Answer) / 87 Slide 1 / 87 Slide 2 / 87 8th Grade Equations with Roots and Radicals 2015-12-17 www.njctl.org Slide 3 / 87 Slide 4 / 87 Table of ontents Radical Expressions ontaining Variables Simplifying Non-Perfect

More information

Algebra I. Slide 1 / 79. Slide 2 / 79. Slide 3 / 79. Equations. Table of Contents Click on a topic to go to that section

Algebra I. Slide 1 / 79. Slide 2 / 79. Slide 3 / 79. Equations. Table of Contents Click on a topic to go to that section Slide 1 / 79 Slide 2 / 79 lgebra I Equations 2015-08-21 www.njctl.org Table of ontents lick on a topic to go to that section. Slide 3 / 79 Equations with the Same Variable on oth Sides Solving Literal

More information

Algebra I. Systems of Linear Equations and Inequalities. Slide 1 / 179. Slide 2 / 179. Slide 3 / 179. Table of Contents

Algebra I. Systems of Linear Equations and Inequalities. Slide 1 / 179. Slide 2 / 179. Slide 3 / 179. Table of Contents Slide 1 / 179 Algebra I Slide 2 / 179 Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Table of Contents Slide 3 / 179 Click on the topic to go to that section 8th Grade Review of

More information

6th Grade. Translating to Equations. Slide 1 / 65 Slide 2 / 65. Slide 4 / 65. Slide 3 / 65. Slide 5 / 65. Slide 6 / 65

6th Grade. Translating to Equations. Slide 1 / 65 Slide 2 / 65. Slide 4 / 65. Slide 3 / 65. Slide 5 / 65. Slide 6 / 65 Slide 1 / 65 Slide 2 / 65 6th Grade Dependent & Independent Variables 15-11-25 www.njctl.org Slide 3 / 65 Slide 4 / 65 Table of Contents Translating to Equations Dependent and Independent Variables Equations

More information

Algebra I System of Linear Equations

Algebra I System of Linear Equations 1 Algebra I System of Linear Equations 2015-11-12 www.njctl.org 2 Table of Contents Click on the topic to go to that section Solving Systems by Graphing Solving Systems by Substitution Solving Systems

More information

Solving and Graphing Linear Inequalities Chapter Questions. 2. Explain the steps to graphing an inequality on a number line.

Solving and Graphing Linear Inequalities Chapter Questions. 2. Explain the steps to graphing an inequality on a number line. Solving and Graphing Linear Inequalities Chapter Questions 1. How do we translate a statement into an inequality? 2. Explain the steps to graphing an inequality on a number line. 3. How is solving an inequality

More information

Module 1-A Linear Inequalities. C. x > 3 D. x < 3 A. 4.4 B. 4.5 C. 4.6 D. 4.7

Module 1-A Linear Inequalities. C. x > 3 D. x < 3 A. 4.4 B. 4.5 C. 4.6 D. 4.7 Name: Date: 1. The inequality 3x + 2 > x + 8 is equivalent to. x > 3 2. x > 3 2 C. x > 3 D. x < 3 2. The inequality 2x > x + 7 is equivalent to. x > 7. x < 7 C. x = 7 D. x > 7 3 3. Which number is not

More information

6th Grade. Equations & Inequalities.

6th Grade. Equations & Inequalities. 1 6th Grade Equations & Inequalities 2015 12 01 www.njctl.org 2 Table of Contents Equations and Identities Tables Determining Solutions of Equations Solving an Equation for a Variable Click on a topic

More information

8th Grade. Slide 1 / 157. Slide 2 / 157. Slide 3 / 157. The Number System and Mathematical Operations Part 2. Table of Contents

8th Grade. Slide 1 / 157. Slide 2 / 157. Slide 3 / 157. The Number System and Mathematical Operations Part 2. Table of Contents Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Table of Contents Slide 3 / 157 Squares of Numbers Greater than 20 Simplifying Perfect

More information

6th Grade. Dependent & Independent Variables

6th Grade. Dependent & Independent Variables Slide 1 / 68 Slide 2 / 68 6th Grade Dependent & Independent Variables 2014-10-28 www.njctl.org Slide 3 / 68 Table of Contents Translating to Equations Dependent and Independent Variables Click on a topic

More information

8th Grade. Two Variable Data. Slide 1 / 122 Slide 2 / 122. Slide 4 / 122. Slide 3 / 122. Slide 6 / 122. Slide 5 / 122. Data.

8th Grade. Two Variable Data. Slide 1 / 122 Slide 2 / 122. Slide 4 / 122. Slide 3 / 122. Slide 6 / 122. Slide 5 / 122. Data. Slide 1 / 122 Slide 2 / 122 8th Grade ata 2015-11-20 www.njctl.org Slide 3 / 122 Slide 4 / 122 Table of ontents click on the topic to go to that section Two Variable ata Line of est Fit etermining the

More information

Study Guide and Review - Chapter 1

Study Guide and Review - Chapter 1 State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The absolute value of a number is always negative. The absolute value of a number is always

More information

Unit 4: Inequalities. Inequality Symbols. Algebraic Inequality. Compound Inequality. Interval Notation

Unit 4: Inequalities. Inequality Symbols. Algebraic Inequality. Compound Inequality. Interval Notation Section 4.1: Linear Inequalities Section 4.2: Solving Linear Inequalities Section 4.3: Solving Inequalities Applications Section 4.4: Compound Inequalities Section 4.5: Absolute Value Equations and Inequalities

More information

Algebra I. Measures of Central Tendency: Mean, Median, Mode & Additional Measures of Data. Slide 1 / 141 Slide 2 / 141. Slide 4 / 141.

Algebra I. Measures of Central Tendency: Mean, Median, Mode & Additional Measures of Data. Slide 1 / 141 Slide 2 / 141. Slide 4 / 141. Slide 1 / 141 Slide 2 / 141 lgebra I ata & Statistical nalysis 2015-11-25 www.njctl.org Slide 3 / 141 Slide 4 / 141 Table of ontents lick on the topic to go to that section Measures of entral Tendency

More information

Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations.

Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations. Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems

More information

6 th Grade - TNREADY REVIEW Q3 Expressions, Equations, Functions, and Inequalities

6 th Grade - TNREADY REVIEW Q3 Expressions, Equations, Functions, and Inequalities 6 th Grade - TNREADY REVIEW Q3 Expressions, Equations, Functions, and Inequalities INSTRUCTIONS: Read through the following notes. Fill in shaded areas and highlight important reminders. Then complete

More information

ALGEBRA 1 SEMESTER 1 INSTRUCTIONAL MATERIALS Courses: Algebra 1 S1 (#2201) and Foundations in Algebra 1 S1 (#7769)

ALGEBRA 1 SEMESTER 1 INSTRUCTIONAL MATERIALS Courses: Algebra 1 S1 (#2201) and Foundations in Algebra 1 S1 (#7769) Multiple Choice: Identify the choice that best completes the statement or answers the question. 1. Ramal goes to the grocery store and buys pounds of apples and pounds of bananas. Apples cost dollars per

More information

Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by David Rennie. Adapted from the Georgia Department of Education Frameworks

Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by David Rennie. Adapted from the Georgia Department of Education Frameworks Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by David Rennie Adapted from the Georgia Department of Education Frameworks Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by

More information

Define the word inequality

Define the word inequality Warm Up: Define the word inequality Agenda: Objective- Students can solve linear inequalities in one variable, including equations with coefficients represented by letters. Define Inequalities One & Two

More information

Algebra 2 Level 2 Summer Packet

Algebra 2 Level 2 Summer Packet Algebra Level Summer Packet This summer packet is for students entering Algebra Level for the Fall of 01. The material contained in this packet represents Algebra 1 skills, procedures and concepts that

More information

Writing and Graphing Inequalities

Writing and Graphing Inequalities .1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle

More information

Topic 1. Solving Equations and Inequalities 1. Solve the following equation

Topic 1. Solving Equations and Inequalities 1. Solve the following equation Topic 1. Solving Equations and Inequalities 1. Solve the following equation Algebraically 2( x 3) = 12 Graphically 2( x 3) = 12 2. Solve the following equations algebraically a. 5w 15 2w = 2(w 5) b. 1

More information

ACCELERATED MATHEMATICS CHAPTER 4 PART II INEQUALITIES TOPICS COVERED:

ACCELERATED MATHEMATICS CHAPTER 4 PART II INEQUALITIES TOPICS COVERED: ACCELERATED MATHEMATICS CHAPTER PART II INEQUALITIES TOPICS COVERED: Solving inequalities with adding and subtracting Solving inequalities with multiplying and dividing Solving two-step inequalities Solving

More information

Inequalities. and Graphing Inequalities 4.3 Solving Inequalities Using. What would you have?

Inequalities. and Graphing Inequalities 4.3 Solving Inequalities Using. What would you have? Inequalities.1 Writing ii in and Graphing Inequalities. Solving Inequalities Using Addition or Subtraction. Solving Inequalities Using Multiplication or Division. Solving Two-Step Inequalities If you reached

More information

Algebra I Notes Linear Inequalities in One Variable and Unit 3 Absolute Value Equations and Inequalities

Algebra I Notes Linear Inequalities in One Variable and Unit 3 Absolute Value Equations and Inequalities PREREQUISITE SKILLS: students must have a clear understanding of signed numbers and their operations students must understand meaning of operations and how they relate to one another students must be able

More information

8th Grade. Equations with Roots and Radicals.

8th Grade. Equations with Roots and Radicals. 1 8th Grade Equations with Roots and Radicals 2015 12 17 www.njctl.org 2 Table of Contents Radical Expressions Containing Variables Click on topic to go to that section. Simplifying Non Perfect Square

More information

SOLVING LINEAR INEQUALITIES

SOLVING LINEAR INEQUALITIES Topic 15: Solving linear inequalities 65 SOLVING LINEAR INEQUALITIES Lesson 15.1 Inequalities on the number line 15.1 OPENER Consider the inequality x > 7. 1. List five numbers that make the inequality

More information

Section 4 Topic 1 Arithmetic Sequences

Section 4 Topic 1 Arithmetic Sequences Section 4 Topic 1 Arithmetic Sequences Let s look at the following sequence of numbers: 3, 8, 13, 18, 23,.... Ø Ø Ø The at the end means that this sequence goes on forever. 3, 8, 13, 18, and 23 are the

More information

7th Grade Math. Expressions & Equations. Table of Contents. 1 Vocab Word. Slide 1 / 301. Slide 2 / 301. Slide 4 / 301. Slide 3 / 301.

7th Grade Math. Expressions & Equations. Table of Contents. 1 Vocab Word. Slide 1 / 301. Slide 2 / 301. Slide 4 / 301. Slide 3 / 301. Slide 1 / 301 Slide 2 / 301 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial

More information

Sample: Do Not Reproduce LF6 STUDENT PAGES LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS. Name Period Date

Sample: Do Not Reproduce LF6 STUDENT PAGES LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS. Name Period Date Name Period Date LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS LF6.1 LF6.2 LF6.3 Introduction to Systems of Linear Equations Understand the definition of a system of linear equations Understand

More information

ALGEBRA 1 FINAL EXAM TOPICS

ALGEBRA 1 FINAL EXAM TOPICS ALGEBRA 1 FINAL EXAM TOPICS Chapter 2 2-1 Writing Equations 2-2 Solving One Step Equations 2-3 Solving Multi-Step Equations 2-4 Solving Equations with the Variable on Each Side 2-5 Solving Equations Involving

More information

Lesson 7: Literal Equations, Inequalities, and Absolute Value

Lesson 7: Literal Equations, Inequalities, and Absolute Value , and Absolute Value In this lesson, we first look at literal equations, which are equations that have more than one variable. Many of the formulas we use in everyday life are literal equations. We then

More information

OTHER METHODS FOR SOLVING SYSTEMS

OTHER METHODS FOR SOLVING SYSTEMS Topic 18: Other methods for solving systems 175 OTHER METHODS FOR SOLVING SYSTEMS Lesson 18.1 The substitution method 18.1 OPENER 1. Evaluate ab + 2c when a = 2, b = 3, and c = 5. 2. Following is a set

More information

Name Per. Keystone Exams Practice Test A.) $300,000 B.) $400,000 C.) $500,000 D.) $600,000

Name Per. Keystone Exams Practice Test A.) $300,000 B.) $400,000 C.) $500,000 D.) $600,000 Name Per Basic Skills Keystone Exams Practice Test 1.) A theme park charges $52 for a day pass and $110 for a week pass. Last month, 4,432 day passes and 979 week passes were sold. Which of the following

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II 1 st Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II 1 st Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

Expressions & Equations Chapter Questions. 6. What are two different ways to solve equations with fractional distributive property?

Expressions & Equations Chapter Questions. 6. What are two different ways to solve equations with fractional distributive property? Expressions & Equations Chapter Questions 1. Explain how distribution can simplify a problem. 2. What are like terms? 3. How do you combine like terms? 4. What are inverse operations? Name them. 5. How

More information

Algebra 1 End-of-Course Assessment Practice Test with Solutions

Algebra 1 End-of-Course Assessment Practice Test with Solutions Algebra 1 End-of-Course Assessment Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit

More information

Final Exam Study Guide

Final Exam Study Guide Algebra 2 Alei - Desert Academy 2011-12 Name: Date: Block: Final Exam Study Guide 1. Which of the properties of real numbers is illustrated below? a + b = b + a 2. Convert 6 yards to inches. 3. How long

More information

Section 2.2 Objectives

Section 2.2 Objectives Section 2.2 Objectives Solve multi-step equations using algebra properties of equality. Solve equations that have no solution and equations that have infinitely many solutions. Solve equations with rational

More information

1 st : Read carefully and underline key words 2 nd : Write a let statement 3 rd : Determine whether to use,,, or 4 th : Write and solve the inequality

1 st : Read carefully and underline key words 2 nd : Write a let statement 3 rd : Determine whether to use,,, or 4 th : Write and solve the inequality Name Period: Represent each of the following as an algebraic inequality. 1) x is at most 30 2) the sum of 5x and 2x is at least 14 3) the product of x and y is less than or equal to 4 4) 5 less than a

More information

CHAPTER 5: ALGEBRA CHAPTER 5 CONTENTS

CHAPTER 5: ALGEBRA CHAPTER 5 CONTENTS CHAPTER 5: ALGEBRA Image from www.coolmath.com CHAPTER 5 CONTENTS 5. Introduction to Algebra 5. Algebraic Properties 5. Distributive Property 5.4 Solving Equations Using the Addition Property of Equality

More information

Equations & Inequalities Chapter Questions. 3. What are two different ways to solve equations with fractional distributive property?

Equations & Inequalities Chapter Questions. 3. What are two different ways to solve equations with fractional distributive property? Equations & Inequalities Chapter Questions 1. What are inverse operations? Name them. 2. How do you solve equations? 3. What are two different ways to solve equations with fractional distributive property?

More information

Systems of Equations Unit Five ONE NONE INFINITE

Systems of Equations Unit Five ONE NONE INFINITE Systems of Equations Unit Five ONE NONE INFINITE Standards: 8.EE.8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables

More information

Foundations for Algebra. Introduction to Algebra I

Foundations for Algebra. Introduction to Algebra I Foundations for Algebra Introduction to Algebra I Variables and Expressions Objective: To write algebraic expressions. Objectives 1. I can write an algebraic expression for addition, subtraction, multiplication,

More information

1Solve systems of. 2Apply Systems of. Then. Why? Now. New Vocabulary system of inequalities

1Solve systems of. 2Apply Systems of. Then. Why? Now. New Vocabulary system of inequalities Systems of Inequalities Then You graphed and solved linear (Lesson 5-6) Now 1Solve systems of linear inequalities by graphing. 2Apply Systems of Why? Jacui is beginning an exercise program that involves

More information

3.1 NOTES Solving Systems of Linear Equations Graphically

3.1 NOTES Solving Systems of Linear Equations Graphically 3.1 NOTES Solving Systems of Linear Equations Graphically A system of two linear equations in two variables x and y consist of two equations of the following form: Ax + By = C Equation 1 Dx + Ey = F Equation

More information

Foundations of Math. Chapter 3 Packet. Table of Contents

Foundations of Math. Chapter 3 Packet. Table of Contents Foundations of Math Chapter 3 Packet Name: Table of Contents Notes #43 Solving Systems by Graphing Pg. 1-4 Notes #44 Solving Systems by Substitution Pg. 5-6 Notes #45 Solving by Graphing & Substitution

More information

Solving Systems of Linear Inequalities Focus on Modeling

Solving Systems of Linear Inequalities Focus on Modeling Name Class 5-6 Date Solving Systems of Linear Inequalities Focus on Modeling Essential question: How can you use systems of linear equations or inequalities to model and solve contextual problems? N-Q.1.1*,

More information

Inequalities Chapter Test

Inequalities Chapter Test Inequalities Chapter Test Part 1: For questions 1-9, circle the answer that best answers the question. 1. Which graph best represents the solution of 8 4x < 4 A. B. C. D. 2. Which of the following inequalities

More information

Looking Ahead to Chapter 4

Looking Ahead to Chapter 4 Looking Ahead to Chapter Focus In Chapter, you will learn about functions and function notation, and you will find the domain and range of a function. You will also learn about real numbers and their properties,

More information

Algebra 1 Keystone Remediation Packet Module 1 Anchor 2

Algebra 1 Keystone Remediation Packet Module 1 Anchor 2 Algebra 1 Keystone Remediation Packet Module 1 Anchor 2 A.1.1.2.1.1 Write, solve, and/or graph linear equations using various methods. A.1.1.2.1.2 Use and/or identify an algebraic property to justify any

More information

1. Write an expression of the third degree that is written with a leading coefficient of five and a constant of ten., find C D.

1. Write an expression of the third degree that is written with a leading coefficient of five and a constant of ten., find C D. 1. Write an expression of the third degree that is written with a leading coefficient of five and a constant of ten. 2 2 2. If C = 4x 7x 9 and D = 5x 7x 3, find C D. 3. At an ice cream shop, the profit,,

More information

8th Grade The Number System and Mathematical Operations Part

8th Grade The Number System and Mathematical Operations Part Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Slide 3 / 157 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect

More information

Pennsylvania. Keystone Exams. Algebra I. Item and Scoring Sampler

Pennsylvania. Keystone Exams. Algebra I. Item and Scoring Sampler Pennsylvania Keystone Exams Algebra I Item and Scoring Sampler 2014 Keystone Algebra I Sampler Table of Contents INFORMATION ABOUT ALGEBRA I Introduction.......................................................................................

More information

ALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6

ALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6 ALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6 FALL 2014 0 1 Algebra 1 Section 6.1 Notes: Graphing Systems of Equations System of Equations: a set of two or more equations with the same variables, graphed in the

More information

Algebra I Final Study Guide

Algebra I Final Study Guide 2011-2012 Algebra I Final Study Guide Short Answer Source: www.cityoforlando.net/public_works/stormwater/rain/rainfall.htm 1. For which one month period was the rate of change in rainfall amounts in Orlando

More information

Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities

Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,

More information

Grade Common Core Math

Grade Common Core Math th 5 Grade Common Core Math Operations and Algebraic Thinking Printable Review Problems Standards Included: -Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with

More information

Sample. Test Booklet. Subject: MA, Grade: HS PSSA 2013 Keystone Algebra 1. - signup at to remove - Student name:

Sample. Test Booklet. Subject: MA, Grade: HS PSSA 2013 Keystone Algebra 1. - signup at   to remove - Student name: Test Booklet Subject: MA, Grade: HS PSSA 2013 Keystone Algebra 1 Student name: Author: Pennsylvania District: Pennsylvania Released Tests Printed: Friday May 31, 2013 1 Which of the following inequalities

More information

3. Find the area for each question below. a. (3x 2)(2x + 5) b. 4. Simplify the expressions below. is equal to 1, what is the value of a?

3. Find the area for each question below. a. (3x 2)(2x + 5) b. 4. Simplify the expressions below. is equal to 1, what is the value of a? Permitted resources: 2018 2019 Algebra 1 Midterm Review FSA Approved calculator Algebra 1 FSA Reference Sheet 1. The expression 13x + 5 represents the number of marbles you have after shopping at the game

More information

Why? Step 3 Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair.

Why? Step 3 Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair. Substitution Then You solved systems of equations by graphing. (Lesson 6-1) Now 1Solve systems of equations by using substitution. 2Solve real-world problems involving systems of equations by using substitution.

More information

Unit 2 Linear Equations and Inequalities

Unit 2 Linear Equations and Inequalities Unit 2 Linear Equations and Inequalities Test Date: Name: By the end of this unit, you will be able to Use rate of change to solve problems Find the slope of a line Model real-world data with linear equations

More information

Grade 6 The Number System & Mathematical Operations

Grade 6 The Number System & Mathematical Operations Slide 1 / 206 Slide 2 / 206 Grade 6 The Number System & Mathematical Operations 2015-10-20 www.njctl.org Slide 3 / 206 Table of Contents Addition, Natural Numbers & Whole Numbers Addition, Subtraction

More information

Lesson 1. Unit 6 Practice Problems. Problem 1. Solution

Lesson 1. Unit 6 Practice Problems. Problem 1. Solution Unit 6 Practice Problems Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson

More information

Practice Test 1 BLACKLINE MASTERS

Practice Test 1 BLACKLINE MASTERS Practice Test 1 BLACKLINE MASTERS Name Date Chapter 1: The Number System Answer the questions that follow. 1. Which of the numbers below is not irrational? A. 5 C. 2 9 B. D. 1.34344344434444 2. Which of

More information

Linear Equations and Inequalities

Linear Equations and Inequalities Unit 2 Linear Equations and Inequalities 9/26/2016 10/21/2016 Name: By the end of this unit, you will be able to Use rate of change to solve problems Find the slope of a line Model real-world data with

More information

HW A) SWBAT identify the properties of operations Create flashcards or a some type of foldable that shows the following properties of operations

HW A) SWBAT identify the properties of operations Create flashcards or a some type of foldable that shows the following properties of operations HW A) SWBAT identify the properties of operations Create flashcards or a some type of foldable that shows the following properties of operations including examples: HW B) SWBAT apply properties of operations

More information

Math Class: Algebra I. Summer Review Packet DUE DATE:

Math Class: Algebra I. Summer Review Packet DUE DATE: Name: 2014-15 Math Class: Algebra I Summer Review Packet DUE DATE: About Algebra I Algebra I teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions

More information

Algebra 1 Unit 6: Linear Inequalities and Absolute Value Guided Notes

Algebra 1 Unit 6: Linear Inequalities and Absolute Value Guided Notes Section 6.1: Solving Inequalities by Addition and Subtraction How do we solve the equation: x 12 = 65? How do we solve the equation: x 12 < 65? Graph the solution: Example 1: 12 y 9 Example 2: q + 23

More information

Expressions and Equations 6.EE.9

Expressions and Equations 6.EE.9 Expressions and Equations 6.EE.9 Teacher Notes Common Core State Standard Expressions and Equations 6.EE Represent and analyze quantitative relationships between dependent and independent variables. 9.

More information

Name ALGEBRA 1 MODULE When factored completely, which is a factor of 12a 2 3a?

Name ALGEBRA 1 MODULE When factored completely, which is a factor of 12a 2 3a? Name ALGEBRA MODULE. When factored completely, which is a factor of 2a 2 3a? a. 2a b. (4x 2 + ) c. 3a d. (4x ) 2. Simplify: a. 4 b. 2 ( x 7) xx ( 4) 2 7x 7 2x 3 c. x 3 d. x 7 x 3 3. A person s hair is

More information

ALGEBRA MIDTERM REVIEW SHEET

ALGEBRA MIDTERM REVIEW SHEET Name Date Part 1 (Multiple Choice): Please show ALL work! ALGEBRA MIDTERM REVIEW SHEET 1) The equations 5x 2y 48 and 3x 2y 32 represent the money collected from school concert ticket sales during two class

More information

Mathematics. Standards Plus. Grade COMMON CORE INTERVENTION SAMPLER

Mathematics. Standards Plus. Grade COMMON CORE INTERVENTION SAMPLER Mathematics Standards Plus COMMON CORE INTERVENTION Grade 7 SAMPLER Standards Plus COMMON CORE INTERVENTION Available for Grades 1-8 Language Arts and Math Standards Plus COMMON CORE INTERVENTION Mathematics

More information

constant matrix Study Guide and Review - Chapter 3

constant matrix Study Guide and Review - Chapter 3 Choose the term from above to complete each sentence. 1. A feasible region that is open and can go on forever is called. unbounded 2. To means to seek the best price or profit using linear programming.

More information

More with Systems of Equations

More with Systems of Equations More with Systems of Equations In 2008, 4.7 million Americans went on a rafting expedition. In Georgia, outfitters run whitewater expeditions for ages 8 and up on the Chattooga River. 12.1 Systems of Equations

More information

CRS SKILL LEVEL DESCRIPTION

CRS SKILL LEVEL DESCRIPTION GRE 501 LESSON/NOTES Period Name CRS SKILL LEVEL DESCRIPTION Level 1 ALL students must GRE 301 Locate points on the number line attain mastery at this level R- XEI 506 Solve first degree inequalities that

More information

Grade 6 Mathematics Unit 4 Expressions and Equations. Topic D Inequalities. Name: Mrs. Archacki

Grade 6 Mathematics Unit 4 Expressions and Equations. Topic D Inequalities. Name: Mrs. Archacki Notes Packet #12 Grade 6 Mathematics Unit 4 Expressions and Equations Topic D Inequalities Name: Mrs. Archacki 1 Topic Objectives By the end of this topic you will be able to write inequalities. graph

More information

Grade 6. The Number System & Mathematical Operations.

Grade 6. The Number System & Mathematical Operations. 1 Grade 6 The Number System & Mathematical Operations 2015 10 20 www.njctl.org 2 Table of Contents Addition, Natural Numbers & Whole Numbers Addition, Subtraction and Integers Multiplication, Division

More information

Name: Date: Period: Notes Day 2 Inequalities Vocabulary & Interval Notation

Name: Date: Period: Notes Day 2 Inequalities Vocabulary & Interval Notation Name: Date: Period: Notes Day 2 Inequalities Vocabulary Interval Notation Interval Notation: Start at the point and end at the point. The smallest number possible is and the largest is. To indicate that

More information

Are You Ready? Write each verbal expression as an algebraic expression more than m 2. r increased by 5

Are You Ready? Write each verbal expression as an algebraic expression more than m 2. r increased by 5 Are You Ready? Write each verbal expression as an algebraic expression. 1. 5 more than m 2. r increased by 5 3. 25 minus q 4. the difference of 20 and t 5. the sum of v and 8 6. the product of 4 and w

More information

ALGEBRA 1. Unit 3 Chapter 6. This book belongs to: Teacher:

ALGEBRA 1. Unit 3 Chapter 6. This book belongs to: Teacher: ALGEBRA 1 Teacher: Unit 3 Chapter 6 This book belongs to: UPDATED FALL 2016 1 2 Algebra 1 Section 6.1 Notes: Graphing Systems of Equations Day 1 Warm-Up 1. Graph y = 3x 1 on a coordinate plane. 2. Check

More information

Algebra 1 S1 (#2201) Foundations in Algebra 1 S1 (#7769)

Algebra 1 S1 (#2201) Foundations in Algebra 1 S1 (#7769) Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following courses: Algebra 1 S1 (#2201) Foundations

More information

2. Find the value of y that makes the equation true. 3. Solve for t. 5(t-3) = 2t

2. Find the value of y that makes the equation true. 3. Solve for t. 5(t-3) = 2t Instructional Week 2: January 11-15 ISTEP + 10 Mathematics Focus Topic: Linear Equations and Inequalities with Real-World Application Paced Standards: AI.L.1: Understand that the steps taken when solving

More information

NC Math 1. Released Items. North Carolina End-of-Course Assessment. Published October 2018

NC Math 1. Released Items. North Carolina End-of-Course Assessment. Published October 2018 Released Items Published October 2018 NC Math 1 North Carolina End-of-Course Assessment Public Schools of North Carolina Department of Public Instruction State Board of Education Division of Accountability

More information

Summary and Vocabulary

Summary and Vocabulary Chapter Chapter Summar and Vocabular Equations involving percents ma be written in the form p q = r, where p is the decimal form of the percent, q is the initial quantit, and r is the resulting quantit.

More information

5.2 Algebraic Properties

5.2 Algebraic Properties 5.2 Algebraic Properties Your friend s birthday is next weekend, and you are sending her a birthday card. As usual, you will put a return address label in the upper left corner of the envelope and a stamp

More information

spring98a Math A Regents Exam Test Sampler spring ) ) 2.5

spring98a Math A Regents Exam Test Sampler spring ) ) 2.5 spring98a For what value of x will 8 and x have the same mean (average) as 27 and 5? ).5 2) 8 3) 24 4) 40 6 Which is a factor of x 2 + 5x 24? ) (x + 4) 2) (x 4) 3) (x + 3) 4) (x 3) 2 If 2x = 4(x + 5),

More information

Cumulative chapters 1-3 review Period: 1. Tell whether each graph represents a function.

Cumulative chapters 1-3 review Period: 1. Tell whether each graph represents a function. Cumulative chapters -3 review Name:_ Period:. Tell whether each graph represents a function. a. b.. Determine whether each function has an absolute maximum or absolute minimum. If the graph has neither

More information

1. Dana used a rule to make a number pattern. Her rule is to multiply by 2. Which number pattern follows Dana s rule?

1. Dana used a rule to make a number pattern. Her rule is to multiply by 2. Which number pattern follows Dana s rule? 4 th Grade Sample test Objective 1.1 1. ana used a rule to make a number pattern. Her rule is to multiply by 2. Which number pattern follows ana s rule? 4, 6, 9, 10, 12 2, 4, 8, 16, 32 5, 7, 9, 11, 13

More information

Name: Class: Date: ID: A

Name: Class: Date: ID: A Name: Class: Date: 8th Grade Advanced Topic III, Linear Equations and Systems of Linear Equations, MA.8.A.1.1, MA.8.1.1.2, MA.8.A.1.3, *MA.8.A.1.4, MA.8.A.1.5, MA.8.A.1.6 Multiple Choice Identify the choice

More information

Linear Relations and Functions

Linear Relations and Functions Linear Relations and Functions Why? You analyzed relations and functions. (Lesson 2-1) Now Identify linear relations and functions. Write linear equations in standard form. New Vocabulary linear relations

More information

Advanced Honors and Honors Integrated Math 1 Summer Packet

Advanced Honors and Honors Integrated Math 1 Summer Packet Advanced Honors and Honors Integrated Math 1 Summer Packet This packet is designed to help you review skills from 8 th grade as well as preview additional skills not learned in 8 th grade that you will

More information

Grade 8 Systems of Linear Equations 8.EE.8a-c

Grade 8 Systems of Linear Equations 8.EE.8a-c THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Systems of Linear Equations 8.EE.8a-c 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK

More information

Foundations of Algebra. Learning Goal 3.1 Algebraic Expressions. a. Identify the: Variables: Coefficients:

Foundations of Algebra. Learning Goal 3.1 Algebraic Expressions. a. Identify the: Variables: Coefficients: Learning Goal 3.1 Algebraic Expressions What you need to know & be able to do 1. Identifying Parts of Algebraic Expressions 3.1 Test Things to remember Identify Parts of an expression Variable Constant

More information

Serena: I don t think that works because if n is 20 and you do 6 less than that you get 20 6 = 14. I think we should write! 6 > 4

Serena: I don t think that works because if n is 20 and you do 6 less than that you get 20 6 = 14. I think we should write! 6 > 4 24 4.6 Taking Sides A Practice Understanding Task Joaquin and Serena work together productively in their math class. They both contribute their thinking and when they disagree, they both give their reasons

More information