Unit 4 Linear Inequalities And Systems — Unit Plan

Title	Assessment
Lesson 1 Representing Situations with Inequalities	Grape Constraints Han has a budget of $25 to buy grapes. Write inequalities to represent the number of pounds of grapes that Han could buy in each situation: Grapes cost $1.99 per pound. Grapes cost $2.49 per pound. Grapes cost $ $c$ per pound. Show Solution Sample response: Let $g$ represent the number of pounds of grapes. $1.99g \leq 25$ $2.49g \leq 25$ $cg \leq 25$
Lesson 2 Solutions to Inequalities in One Variable	Seeking Solutions Which graph correctly shows the solution to the inequality $\dfrac{7x-3}9 \geq 8-2x$ ? Show or explain your reasoning. A B C D Show Solution Graph C. Sample reasoning: I tested 2, 3, and 4 for the values of $x$ , and 3 and 4 make the inequality true. I solved a related equation and then tested a couple of values on either side of the solution.
Lesson 3 Writing and Solving Inequalities in One Variable	How Many Hours of Work? Lin’s job pays $8.25 an hour plus $10 of transportation allowance each week. She has to work at least 5 hours a week to keep the job, and can earn up to $175 per week (including the allowance). Represent this situation mathematically. If you use variables, specify what each one means. How many hours per week can Lin work? Explain or show your reasoning. Show Solution Sample response: $8.25h + 10 \leq 175$ and $h \geq 5$ , where $h$ represents the number of hours Lin works in a week. At least 5 hours and at most 20 hours (or $5\leq h \leq 20$ ). Sample reasoning: The maximum amount she could earn, not including the transportation allowance, is $165. That amount is equal to 20 hours of work ( $\dfrac {165}{8.25} = 20$ ).
Section A Check Section A Checkpoint	Problem 1 Solve the inequality, and use the number line to represent the solution. $3\left( 2x - \frac{2}{3} \right) < x+1$ Show Solution $x < \frac{3}{5}$ (or equivalent) Problem 2 In a cooperative video game, the gold collected on a mission is shared evenly between the 2 players. To afford the next upgrade, Andre’s character needs 20 gold. Andre only has 6 gold right now. Let $g$ represent the amount of gold the 2 players will collect on the next mission. Write an inequality that represents the condition that Andre gets enough gold to buy the upgrade. Show Solution $\frac{1}{2}g + 6 \geq 20$ or equivalent

Lesson 4 Graphing Linear Inequalities in Two Variables (Part 1)	Pick a Graph The line in each graph represents $y=2x$ . Which graph represents $2x>y$ ? A Inequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Dashed line passes through negative 4 comma negative 8, 0 comma 0, and 4 comma 8. The region above the dashed line is shaded. B Inequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Solid line passes through negative 4 comma negative 8, 0 comma 0, and 4 comma 8. The region above the solid line is shaded. C Inequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Dashed line passes through negative 4 comma negative 8, 0 comma 0, and 4 comma 8. The region below the dashed line is shaded. D Inequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Solid line passes through negative 4 comma negative 8, 0 comma 0, and 4 comma 8. The region below the solid line is shaded. Explain your reasons for choosing that graph. Show Solution Graph C Sample response: I substituted the coordinates of a few points above the line into the inequality and found that they are all not solutions. The point $(0,0)$ , which is on the line, is also not a solution. I concluded that the points on and above the line are not solutions, and the region below the line represents the solutions.
Lesson 5 Graphing Linear Inequalities in Two Variables (Part 2)	A Weekend of Games To raise money for after-school programs at an elementary school, a group of parents is holding a weekend of games in a community center. They charge $8 per person for entry into the event. The group would like to earn at least $600, after paying for the cost of renting the space, which is $40 an hour. If $x$ represents the number of entry tickets sold and $y$ the hours of space rental, which inequality represents the constraints in the situation? $8x-40y<600$ $8x-40y\leq600$ $8x-40y>600$ $8x-40y\geq600$ The line is the graph of $8x-40y=600$ . Select all points whose $(x,y)$ values represent the group reaching its fundraising goal. Explain or show your reasoning. Graph of a line, origin O. Horizontal axis, number of tickets sold, scale is 0 to 250, by 50’s. Vertical axis, hours of rental, scale is 0 to 50, by 10’s. Line starts at 75 comma 0 and passes through 125 comma 10 and 180 comma 21. Point A, 100 comma 10, point B 180 comma 21, and point C 200 comma 5 are identified. Complete the graph so that it represents solutions to an inequality that represents this situation. (Be clear about whether you want to use a solid or dashed line.) Show Solution D Points B and C. Sample explanation: When the coordinates of B and C are substituted for the $x$ and $y$ in the equation, they result in a number that is at least 600.
Lesson 6 Solving Problems with Inequalities in Two Variables	The Band Played On A band is playing at an auditorium with floor seats and balcony seats. The band wants to sell the floor tickets for $15 each and balcony tickets for $12 each. They want to make at least $3,000 in ticket sales. How much money will they collect for selling $x$ floor tickets? How much money will they collect for selling $y$ balcony tickets? Write an inequality whose solutions are the number of floor and balcony tickets sold if they make at least $3,000 in ticket sales. Use technology to graph the solutions to your inequality, and sketch the graph. Show Solution $15x$ $12y$ $15x+12y \geq 3,000$ See graph.
Section B Check Section B Checkpoint	Problem 1 Elena’s family has budgeted $200 for the year for streaming services. One service charges $15.50 per month, and another charges $10 per month. Write an inequality whose solutions are the number of months subscribed to the two services if they work within the budget. Use $x$ for the number of months they subscribe to the more expensive service and $y$ for the number of months they subscribe to the less expensive service. Show Solution $15.50x + 10y \leq 200$ Problem 2 The graph represents the solution to an inequality. Write one pair of values that satisfy the inequality. $x=\\y=$ Does a point on the boundary line represent a solution to the inequality? Explain your reasoning. Show Solution Sample response: $x = 5, y = 11$ No. It is a dashed line, which indicates that points on the lines are not solutions.

Lesson 7 Solutions to Systems of Linear Inequalities in Two Variables	Oh Good, Another Riddle Here is another riddle: The sum of two numbers is less than 2. If we subtract the second number from the first, the difference is greater than 1. What are the two numbers? The riddle can be represented by a system of inequalities. Write an inequality for each statement. These graphs represent the inequalities in the system. Which graph represents which inequality? A graph of two intersecting inequalities on a coordinate plane, origin O. Each axis from negative 2 to 3, by 1’s. The first dashed line starts below x axis and left of y axis, goes through 0 comma negative 1, 1 comma 0, and 3 comma 2. The region below the dashed line is shaded. Second line starts on negative 2 comma 4, goes through 1 comma 1, and 3 comma negative 1. The region below the dashed line is shaded. Name a possible solution to the riddle. Explain or show how you know. Show Solution $\begin {cases} x+y<2\\ x-y>1 \end {cases}$ The region with solid blue shading represents the first clue. The region with line shading represents the second clue. Sample response: $(1.5,0)$ . The sum is 1.5, which is less than 2. The difference is 1.5, which is greater than 1. The point $(1.5,0)$ is located in the region where the two shaded regions overlap, which means it is a solution to both inequalities in the system.
Lesson 8 Solving Problems with Systems of Linear Inequalities in Two Variables	Widgets and Zurls A factory produces widgets and zurls. The combined number of widgets and zurls made each day cannot be more than 12. The maximum number of widgets the factory can produce in a day is 4. Let $x$ be the number of widgets and $y$ the number of zurls. Select all the inequalities that represent this situation. $x < 4$ $x \le 4$ $x>4$ $x+y>12$ $x+y \le 12$ Here are graphs of $x=4$ and $x+y=12$ . Complete the graphs (by shading regions and adjusting line types as needed) to show all the allowable numbers of widgets and zurls that the factory can produce in one day. Does each ordered pair represent an allowable combination of widgets and zurls produced in one day? $(4,5)$ $(11,1)$ $(4,12)$ $(3,9)$ Show Solution $x \le 4$ and $x+y \le 12$ See graph. Yes, no, no, yes
Lesson 9 Modeling with Systems of Inequalities in Two Variables	No cool-down
Section C Check Section C Checkpoint	Problem 1 Draw a graph of a system of inequalities that has no solution. Show Solution Sample response: Problem 2 To have a valid password on a website, the password must have at least 2 letters, at least 1 number, and be less than 8 characters (letters or numbers) in length. Write 3 inequalities to represent the constraints. Make sure to include what each variable represents. Would the password j8675309 be a solution to the inequalities you wrote? Explain your reasoning. Graph the solution region for the system of 3 inequalities. Show Solution Sample response: $x \geq 2, y \geq 1, x + y < 8$ where $x$ is the number of letters and $y$ is the number of numbers in the password. No. Sample reasoning: There is only 1 letter and there are 8 characters, so $x < 2$ and $x + y = 8$ , which do not fit the constraints.