Unit 4 Plan - Algebra 1

Title	Takeaways	Student Summary	Assessment
Lesson 1 Representing Situations with Inequalities	—	We have used equations and the equal sign to represent relationships and constraints in various situations. Not all relationships and constraints involve equality, however. In some situations, one quantity is, or needs to be, greater than or less than another. To describe these situations, we can use inequalities and symbols such as $<, \leq, >$ , or $\geq$ . When working with inequalities, it helps to remember what the symbol means, in words. For example: $100 < a$ means “100 is less than $a$ .” $y \le 55$ means “ $y$ is less than or equal to 55,” or " $y$ is not more than 55." $20 > 18$ means “20 is greater than 18.” $t \ge 40$ means “ $t$ is greater than or equal to 40,” or " $t$ is at least 40." These inequalities are fairly straightforward. Each inequality states the relationship between two numbers ( $20>18$ ), or it describes the limit or boundary of a quantity in terms of a number ( $100<a></a></span>). </p> <p>Inequalities can also express relationships or constraints that are more complex. Here are some examples:</p> <div class="imgrid"> <div class="g--row"> <div class="g--column g--content g--three-fourth"> <ul> <li>The area of a rectangle, <span class="math">\(A$ , with a length of 4 meters and a width of $w$ meters is no more than 100 square meters. $A \leq 100$ $4w\leq100$ To cover all the expenses of a musical production each week, the number of weekday tickets sold, $d$ , and the number of weekend tickets sold, $s$ , must be greater than 4,000. $d + s>4,000$ Elena would like the number of hours she works in a week, $h$ , to be more than 5 but no more than 20. $h>5$ $h \leq 20$ The total cost, $T$ , of buying $a$ adult shirts and $c$ child shirts must be less than 150. Adult shirts are $12 each, and child shirts are $7 each. $T<150$ $12a + 7c < 150$ In upcoming lessons, we’ll use inequalities to help us solve problems.	Grape Constraints (1 problem) 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	—	The equation $\frac12 t = 10$ is an equation in one variable. Its solution is any value of $t$ that makes the equation true. Only $t=20$ meets that requirement, so 20 is the only solution. The inequality $\frac12t >10$ is an inequality in one variable. Any value of $t$ that makes the inequality true is a solution. For instance, 30 and 48 are both solutions because substituting these values for $t$ produces true inequalities. $\frac12(30) >10$ is true, as is $\frac12(48) >10$ . Because the inequality has a range of values that make it true, we sometimes refer to all the solutions as the solution set. One way to find the solutions to an inequality is by reasoning. For example, to find the solution to $2p<8$ , we can reason that if 2 times a value is less than 8, then that value must be less than 4. So a solution to $2p<8$ is any value of $p$ that is less than 4. Another way to find the solutions to $2p<8$ is to solve the related equation $2p=8$ . In this case, dividing each side of the equation by 2 gives $p=4$ . This point, where $p$ is 4, is the boundary of the solution to the inequality. To find out the range of values that make the inequality true, we can try values less than and greater than 4 in our inequality and see which ones make a true statement. Let's try some values less than 4: If $p=3$ , the inequality is $2(3) <8$ or $6 < 8$ , which is true. If $p=\text-1$ , the inequality is $2(\text-1) < 8$ or $\text-2 <8$ , which is also true. Let's try values greater than 4: If $p=5$ , the inequality is $2(5)<8$ or $10<8$ , which is false. If $p=12$ , the inequality is $2(12) <8$ or $24<8$ , which is also false. In general, the inequality is false when $p$ is greater than or equal to 4 and true when $p$ is less than 4. We can represent the solution set to an inequality by writing an inequality, $p<4$ , or by graphing on a number line. The ray pointing to the left represents all values less than 4.	Seeking Solutions (1 problem) 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	—	Writing and solving inequalities can help us make sense of the constraints in a situation and solve problems. Let's look at an example. Clare would like to buy a video game system that costs $130 and to have some extra money for games. She has saved $48 so far and plans on saving $5 of her allowance each week. How many weeks, $w$ , will it be until she has enough money to buy the system and have some extra money remaining? To represent the constraints, we can write $48 + 5w \geq 130$ . Let’s reason about the solutions: Because Clare has $48 already and needs to have at least $130 to afford the game system, she needs to save at least $82 more. If she saves $5 each week, it will take at least $\frac{82}{5}$ weeks to reach $82. $\frac{82}{5}$ is 16.4. Any time shorter than 16.4 weeks won't allow her to save enough. Assuming she saves $5 at the end of each week (instead of saving smaller amounts throughout a week), it will be at least 17 weeks before she can afford the game system. We can also solve by writing and solving a related equation to find the boundary value for $w$ , and then determine whether the solutions are less than or greater than that value. $\begin{aligned} 48 + 5w &= 130\\ 5w & = 82\\ w &=\frac{82}{5} \\w&=16.4 \end{aligned}$ Substituting 16.4 for $w$ in the original inequality gives a true statement. (When $w=16.4$ , we get $130 \geq 130$ .) Substituting a value greater than 16.4 for $w$ also gives a true statement. (When $w = 17$ , we get $133\geq130$ .) Substituting a value less than 16.4 for $w$ gives a false statement. (When $w=16$ , we get $128\geq130$ .) The solution set is therefore $w \geq 16.4$ . Sometimes the structure of an inequality can help us see whether the solutions are less than or greater than a boundary value. For example, to find the solutions to $3x > 8x$ , we can solve the equation $3x = 8x$ , which gives us $x = 0$ . Then, instead of testing values on either side of 0, we could reason as follows about the inequality: If $x$ is a positive value, then $3x$ would be less than $8x$ . For $3x$ to be greater than $8x$ , $x$ must include negative values. For the solutions to include negative values, they must be less than 0, so the solution set would be $x < 0$ .	How Many Hours of Work? (1 problem) 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

Lesson 4 Graphing Linear Inequalities in Two Variables (Part 1)	—	The equation $x+y = 7$ is an equation in two variables. Its solution is any pair of $x$ and $y$ whose sum is 7. The pairs $x=0, y=7$ and $x =\text5, y= 2$ are two examples. We can represent all the solutions to $x+y = 7$ by graphing the equation on a coordinate plane. The graph is a line. All the points on the line are solutions to $x+y = 7$ . The inequality $x+y \leq 7$ is an inequality in two variables. Its solution is any pair of $x$ and $y$ whose sum is 7 or less than 7. This means it includes all the pairs that are solutions to the equation $x+y=7$ , but also many other pairs of $x$ and $y$ that add up to a value less than 7. The pairs $x=4, y=\text-7$ and $x=\text-6, y=0$ are two examples. On a coordinate plane, the solution to $x+y \leq 7$ includes the line that represents $x+y=7$ . If we plot a few other $(x,y)$ pairs that make the inequality true, such as $(4, \text-7)$ and $(\text-6,0)$ , we see that these points fall on one side of the line. (In contrast, $(x,y)$ pairs that make the inequality false fall on the other side of the line.) We can shade that region on one side of the line to indicate that all points in it are solutions. Inequality graphed on a coordinate plane, origin O. Each axis from negative 8 to 10, by 2’s. Solid line goes through 0 comma 7 and 5 comma 2. The region below the solid line is shaded. The points negative 6 comma 0 and 4 comma negative 7 are labeled. What about the inequality $x+y <7$ ? The solution is any pair of $x$ and $y$ whose sum is less than 7. This means pairs like $x=0, y=7$ and $x =5, y=2$ are not solutions. On a coordinate plane, the solution does not include points on the line that represent $x+y=7$ (because those points are $x$ and $y$ pairs whose sum is 7). To exclude points on that boundary line, we can use a dashed line. All points below that line are $(x,y)$ pairs that make $x+y<7$ true. The region on that side of the line can be shaded to show that it contains the solutions. Inequality graphed on a coordinate plane, origin O. Each axis from negative 8 to 10, by 2’s. Dashed line goes through 0 comma 7 and 7 comma 0. The region below the dashed line is shaded.	Pick a Graph (1 problem) 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 6 Solving Problems with Inequalities in Two Variables	—	Suppose we want to find the solution to $x - y > 5$ . We can start by graphing the related equation $x - y = 5$ . When identifying the solution region, it is important not to assume that the solution will be above the line because of a “>” symbol or below the line because of a “<” symbol. Instead, test the points on the line and on either side of the line, and see if they are solutions. Graph of a line, origin O, with grid. X axis from negative 2 to 12 by 2’s. Y axis from negative 6 to 8 by 2’s. Line passes through approximately 1 comma negative 4, 5 comma 0, and 12 comma 7. For $x-y>5$ , points on the line and above the line are not solutions to the inequality because the $(x,y)$ pairs make the inequality false. Points that are below the lines are solutions, so we can shade that lower region. Graphing technology can help us graph the solution to an inequality in two variables. Many graphing tools allow us to enter inequalities such as $x-y >5$ and will show the solution region, as shown here. Some tools, however, may require the inequalities to be in slope-intercept form or another form before displaying the solution region. Be sure to learn how to use the graphing technology available in your classroom. Inequality graphed on a coordinate plane. X axis from 0 to 10, by 5’s. Y axis from negative 5 to 10, by 5’s. Dashed line goes through 0 comma negative 5, 5 comma 0, and 10 comma 5. The region below the dashed line is shaded. Although graphing using technology is efficient, we still need to analyze the graph with care. Here are some things to consider: The graphing window. If the graphing window is too small, we may not be able to really see the solution region or the boundary line, as shown here. The meaning of solution points in the situation. For example, if $x$ and $y$ represent the lengths of two sides of a rectangle, then only positive values of $x$ and $y$ (or points in the first quadrant) make sense in the situation. Inequality graphed on a coordinate plane. Each axis from negative 2 to 4, by 2’s. Dashed line goes through 3 comma negative 2, 3 point 5 comma negative 1 point 5, and 4 comma negative 1. The region below the dashed line is shaded.	The Band Played On (1 problem) 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

Lesson 7 Solutions to Systems of Linear Inequalities in Two Variables	—	In this lesson, two linear inequalities in two variables represent the constraints in a situation. Each pair of inequalities forms a system of inequalities. A solution to a system of inequalities is any $(x,y)$ pair that makes both inequalities true, or any pair of values that simultaneously meet both constraints in the situation. The solution to the system is often best represented by a region on a graph. Suppose there are two numbers, $x$ and $y$ , and there are two things we know about them. The value of one number is more than double the value of the other. The sum of the two numbers is less than 10. We can represent these constraints with a system of inequalities. $\begin {cases} y > 2x\\ x+y <10 \end {cases}$ There are many possible pairs of numbers that meet the first constraint, for example: 1 and 3, or 4 and 9. The same can be said about the second constraint, for example: 1 and 3, or 2.4 and 7.5. The pair $x=1$ and $y=3$ meets both constraints, so it is a solution to the system. The pair $x=4$ and $y=9$ meets the first constraint but not the second ( $9 >2(4)$ is a true statement, but $4+9<10$ is not true.) Remember that graphing is a great way to show all the possible solutions to an inequality, so let’s graph the solution region for each inequality. A graph of an inequality on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. Dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded. A graph of an inequality on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. A dashed line starts on the y axis at 10, goes through 5 comma 5, and ends on x axis at 10. The region below the dashed line is shaded. Because we are looking for a pair of numbers that meet both constraints or make both inequalities true at the same time, we want to find points that are in the solution regions of both graphs. To do that, we can graph both inequalities on the same coordinate plane. The solution set to the system of inequalities is represented by the region where the two graphs overlap. A graph of two intersecting inequalities on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. The first dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded. Second line at on the y axis at 10, goes through 5 comma 5, end on x axis at 10. The region below the dashed line is shaded.	Oh Good, Another Riddle (1 problem) 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 9 Modeling with Systems of Inequalities in Two Variables	—	—	No cool-down
Section C Check Section C Checkpoint
Unit 4 Assessment End-of-Unit Assessment	Problem 1 Consider the inequality $\dfrac{x+3}6 > \dfrac {x}{4} +1$ . Which value of $x$ is a solution to the inequality? A. $x=\text-10$ ✓ B. $x=\text-6$ C. $x=6$ D. $x=10$ Show Solution A Problem 2 The number of fish in a pond is eight more than the number of frogs. The total number of fish and frogs in the pond is at least 20. If $x$ represents the number of frogs, which inequality can be used to represent this situation? A. $x + 8x \geq 20$ B. $2x + 8 \geq 20$ ✓ C. $x + 8x \leq 20$ D. $2x + 8 \leq 20$ Show Solution B Problem 3 Which graph represents the solution to this system of inequalities? $\begin {cases} 3x-5y \leq 15\\ y > \text- \frac 23 x+1 \end {cases}$ A A graph of two intersecting inequalities on a coordinate plane, origin O. Horizontal x axis from negative 8 to 11, by 1's. Vertical y axis from negative 6 to 5, by 1's. Solid line starts at negative 5 comma negative 6, passes through 0 comma negative 3, 5 comma 0, and 10 comma 3. The region below the solid line is shaded. Dashed line starts above the x axis and to the left of the y axis, and passes through 0 comma 1, 3 comma negative 1, and 9 comma negative 5. The area below the dotted line is shaded. B A graph of two intersecting inequalities on a coordinate plane, origin O. Horizontal x axis from negative 8 to 11, by 1's. Vertical y axis from negative 6 to 5, by 1's. Solid line starts at negative 5 comma negative 6, passes through 0 comma negative 3, 5 comma 0, and 10 comma 3. The region above the solid line is shaded. Dashed line passes through 0 comma 1, 3 comma negative 1, and 9 comma negative 5. The area above the dotted line is shaded. C A graph of two intersecting inequalities on a coordinate plane, origin O. Horizontal x axis from negative 8 to 11, by 1's. Vertical y axis from negative 6 to 5, by 1's. Solid line starts at negative 5 comma negative 6, passes through 0 comma negative 3, 5 comma 0, and 10 comma 3. The region above the solid line is shaded. Dashed line starts above the x axis and to the left of the y axis, and passes through 0 comma 1, 3 comma negative 1, and 9 comma negative 5. The area above the dotted line is shaded. D A graph of two intersecting inequalities on a coordinate plane, origin O. Horizontal x axis from negative 8 to 11, by 1's. Vertical y axis from negative 6 to 5, by 1's. Solid line starts at negative 5 comma negative 6, passes through 0 comma negative 3, 5 comma 0, and 10 comma 3. The region above the solid line is shaded. Dashed line starts above the x axis and to the left of the y axis, and passes through 0 comma 1, 3 comma negative 1, and 9 comma negative 5. The area below the dotted line is shaded. A. Graph A B. Graph B ✓ C. Graph C D. Graph D Show Solution B Problem 4 Graph the solution to the inequality $4x+5y<20$ . Blank x y coordinate plane with grid and origin labeled O. Both axes labeled from negative 6 to 5 by 1's. Show Solution Key features: boundary line is dashed (strict inequality), intercepts at (0,4) and (5,0), region below the line is shaded. Problem 5 A hairstylist charges $15 for an adult haircut and $9 for a child haircut. She wants to earn at least $360 dollars and cut a maximum of 30 haircuts this week. The graphs represent the hairstylist's constraints. Two inequalities graphed on a coordinate plane, origin O, scale from 0 to 45 on both axes. Horizontal axis, number of adult haircuts. Vertical axis, number of child haircuts. A solid line starts on vertical axis at 40, goes through 15 comma 15 and ends on horizontal axis at 24. The region above the solid line is shaded. Another solid line starts on the vertical axis at 30, goes through 5 comma 25, 10 comma 20, 20 comma 10, and ends on the horizontal axis at 30. The region below the solid line is shaded. List two points that could represent the numbers of adult and child haircuts that meet the hairstylist's goals. Show Solution Sample response: $(25,0)$ , $(25,2)$ . Any two points with nonnegative integer coordinates in the overlapping shaded region are acceptable. Problem 6 A jewelry artist is selling necklaces at an art fair. It costs $135 to rent a booth at the fair. The cost of materials for each necklace is $4.50. The artist is selling the necklaces at $12 each. The inequality $12n > 135 + 4.50n$ represents the situation in which the artist makes a profit. Will the artist make a profit if she sells 15 necklaces? Show how you know. Write an equivalent inequality with $n$ by itself on one side. Show your reasoning. Show Solution No. $12(15)=180$ but $135+4.50(15)=202.50$ . Since $180 < 202.50$ , the artist is not making a profit. $n>18$ . Reasoning: $12n > 135 + 4.50n \Rightarrow 7.50n > 135 \Rightarrow n > 18$ . Problem 7 A student has started a lawn care business. He charges $15 per hour to mow lawns and $20 per hour for gardening. Because he is still in school, he is allowed to work for at most 20 hours per week. His goal is to make at least $300 per week. Create a system of equations or inequalities that models the situation. Define the variables that you use. The graph shows one of the relevant equations in this situation. Draw the graph of the other relevant equation. A blank coordinate grid with origin 0. Horizontal axis 0 to 24, by 2's. Vertical axis from 0 to 24, by 2's. Show all the points that represent the number of hours the student can work at each job and meet his goal. Show Solution Where $x$ = hours mowing and $y$ = hours gardening: $\begin{cases} 15x+20y \geq 300 \\ x+y \leq 20 \end{cases}$ The region where the two graphs overlap represents the number of hours the student can work and meet his goal. Problem 8 Which graph is the solution to the inequality $6.4 - 4x \geq -2.8$ ? A. A. B. B. C. C. D. D. ✓ Show Solution D Problem 9 Graph the system of inequalities on the set of axes below: $\begin{cases} y > 3x - 4 \\ x + 2y \leq 6 \end{cases}$ Label the solution set S. Show Solution Dashed line through $(0, -4)$ and $(\frac{4}{3}, 0)$ with slope 3, region above shaded. Solid line through $(0, 3)$ and $(6, 0)$ with slope $-\frac{1}{2}$ , region below shaded. The overlapping region is labeled S.

Representing Situations with Inequalities

—

We have used equations and the equal sign to represent relationships and constraints in various situations. Not all relationships and constraints involve equality, however.

In some situations, one quantity is, or needs to be, greater than or less than another. To describe these situations, we can use inequalities and symbols such as $<, \leq, >$ , or $\geq$ .

When working with inequalities, it helps to remember what the symbol means, in words. For example:

$100 < a$ means “100 is less than $a$ .”
$y \le 55$ means “ $y$ is less than or equal to 55,” or " $y$ is not more than 55."
$20 > 18$ means “20 is greater than 18.”
$t \ge 40$ means “ $t$ is greater than or equal to 40,” or " $t$ is at least 40."

These inequalities are fairly straightforward. Each inequality states the relationship between two numbers ( $20>18$ ), or it describes the limit or boundary of a quantity in terms of a number ( $100<a></a></span>). </p> <p>Inequalities can also express relationships or constraints that are more complex. Here are some examples:</p> <div class="imgrid"> <div class="g--row"> <div class="g--column g--content g--three-fourth"> <ul> <li>The area of a rectangle, <span class="math">\(A$ , with a length of 4 meters and a width of $w$ meters is no more than 100 square meters.

$A \leq 100$
$4w\leq100$

To cover all the expenses of a musical production each week, the number of weekday tickets sold, $d$ , and the number of weekend tickets sold, $s$ , must be greater than 4,000.

$d + s>4,000$

Elena would like the number of hours she works in a week, $h$ , to be more than 5 but no more than 20.

$h>5$
$h \leq 20$

The total cost, $T$ , of buying $a$ adult shirts and $c$ child shirts must be less than 150. Adult shirts are $12 each, and child shirts are $7 each.

$T<150$
$12a + 7c < 150$

In upcoming lessons, we’ll use inequalities to help us solve problems.

Grape Constraints (1 problem)

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

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The equation $\frac12 t = 10$ is an equation in one variable. Its solution is any value of $t$ that makes the equation true. Only $t=20$ meets that requirement, so 20 is the only solution.

The inequality $\frac12t >10$ is an inequality in one variable. Any value of $t$ that makes the inequality true is a solution. For instance, 30 and 48 are both solutions because substituting these values for $t$ produces true inequalities. $\frac12(30) >10$ is true, as is $\frac12(48) >10$ . Because the inequality has a range of values that make it true, we sometimes refer to all the solutions as the solution set.

One way to find the solutions to an inequality is by reasoning. For example, to find the solution to $2p<8$ , we can reason that if 2 times a value is less than 8, then that value must be less than 4. So a solution to $2p<8$ is any value of $p$ that is less than 4.

Another way to find the solutions to $2p<8$ is to solve the related equation $2p=8$ . In this case, dividing each side of the equation by 2 gives $p=4$ . This point, where $p$ is 4, is the boundary of the solution to the inequality.

To find out the range of values that make the inequality true, we can try values less than and greater than 4 in our inequality and see which ones make a true statement.

Let's try some values less than 4:

If $p=3$ , the inequality is $2(3) <8$ or $6 < 8$ , which is true.
If $p=\text-1$ , the inequality is $2(\text-1) < 8$ or $\text-2 <8$ , which is also true.

Let's try values greater than 4:

If $p=5$ , the inequality is $2(5)<8$ or $10<8$ , which is false.
If $p=12$ , the inequality is $2(12) <8$ or $24<8$ , which is also false.

In general, the inequality is false when $p$ is greater than or equal to 4 and true when $p$ is less than 4.

We can represent the solution set to an inequality by writing an inequality, $p<4$ , or by graphing on a number line. The ray pointing to the left represents all values less than 4.

<p>Inequality graphed on a number line. Numbers from negative 6 to 6, by 1’s. At 4, open circle with line extending to the left.</p>

Seeking Solutions (1 problem)

Which graph correctly shows the solution to the inequality $\dfrac{7x-3}9 \geq 8-2x$ ? Show or explain your reasoning.

<p>Inequality graphed on a number line. Numbers from negative 3 to 10, by 1's. At 3, open circle with line extending to the right.</p> — A

<p>Inequality graphed on a number line. Numbers from negative 3 to 10. At 3, open circle with line extending to the left.</p> — B

<p>Inequality graphed on a number line. Numbers from negative 3 to 10. At 3, closed circle with line extending to the right.</p> — C

<p>Inequality graphed on a number line. Numbers from negative 3 to 10. At 3, closed circle with line extending to the left.<br>
</p> — 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

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Writing and solving inequalities can help us make sense of the constraints in a situation and solve problems. Let's look at an example.

Clare would like to buy a video game system that costs $130 and to have some extra money for games. She has saved $48 so far and plans on saving $5 of her allowance each week. How many weeks, $w$ , will it be until she has enough money to buy the system and have some extra money remaining? To represent the constraints, we can write $48 + 5w \geq 130$ . Let’s reason about the solutions:

Because Clare has $48 already and needs to have at least $130 to afford the game system, she needs to save at least $82 more.
If she saves $5 each week, it will take at least $\frac{82}{5}$ weeks to reach $82.
$\frac{82}{5}$ is 16.4. Any time shorter than 16.4 weeks won't allow her to save enough.
Assuming she saves $5 at the end of each week (instead of saving smaller amounts throughout a week), it will be at least 17 weeks before she can afford the game system.

We can also solve by writing and solving a related equation to find the boundary value for $w$ , and then determine whether the solutions are less than or greater than that value.

$\begin{aligned} 48 + 5w &= 130\\ 5w & = 82\\ w &=\frac{82}{5} \\w&=16.4 \end{aligned}$

Substituting 16.4 for $w$ in the original inequality gives a true statement. (When $w=16.4$ , we get $130 \geq 130$ .)
Substituting a value greater than 16.4 for $w$ also gives a true statement. (When $w = 17$ , we get $133\geq130$ .)
Substituting a value less than 16.4 for $w$ gives a false statement. (When $w=16$ , we get $128\geq130$ .)
The solution set is therefore $w \geq 16.4$ .

Sometimes the structure of an inequality can help us see whether the solutions are less than or greater than a boundary value. For example, to find the solutions to $3x > 8x$ , we can solve the equation $3x = 8x$ , which gives us $x = 0$ . Then, instead of testing values on either side of 0, we could reason as follows about the inequality:

If $x$ is a positive value, then $3x$ would be less than $8x$ .
For $3x$ to be greater than $8x$ , $x$ must include negative values.
For the solutions to include negative values, they must be less than 0, so the solution set would be $x < 0$ .

How Many Hours of Work? (1 problem)

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

Lesson 4

Graphing Linear Inequalities in Two Variables (Part 1)

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The equation $x+y = 7$ is an equation in two variables. Its solution is any pair of $x$ and $y$ whose sum is 7. The pairs $x=0, y=7$ and $x =\text5, y= 2$ are two examples.

We can represent all the solutions to $x+y = 7$ by graphing the equation on a coordinate plane.

The graph is a line. All the points on the line are solutions to $x+y = 7$ .

<p>Graph of a line, origin O, with grid. Scale is negative 8 to 10, by 2’s on both axes. Line passes through 0 comma 7 and 5 comma 2.</p>

The inequality $x+y \leq 7$ is an inequality in two variables. Its solution is any pair of $x$ and $y$ whose sum is 7 or less than 7.

This means it includes all the pairs that are solutions to the equation $x+y=7$ , but also many other pairs of $x$ and $y$ that add up to a value less than 7. The pairs $x=4, y=\text-7$ and $x=\text-6, y=0$ are two examples.

On a coordinate plane, the solution to $x+y \leq 7$ includes the line that represents $x+y=7$ . If we plot a few other $(x,y)$ pairs that make the inequality true, such as $(4, \text-7)$ and $(\text-6,0)$ , we see that these points fall on one side of the line. (In contrast, $(x,y)$ pairs that make the inequality false fall on the other side of the line.)

We can shade that region on one side of the line to indicate that all points in it are solutions.

What about the inequality $x+y <7$ ?

The solution is any pair of $x$ and $y$ whose sum is less than 7. This means pairs like $x=0, y=7$ and $x =5, y=2$ are not solutions.

On a coordinate plane, the solution does not include points on the line that represent $x+y=7$ (because those points are $x$ and $y$ pairs whose sum is 7).

To exclude points on that boundary line, we can use a dashed line.

All points below that line are $(x,y)$ pairs that make $x+y<7$ true. The region on that side of the line can be shaded to show that it contains the solutions.

Pick a Graph (1 problem)

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 6

Solving Problems with Inequalities in Two Variables

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Suppose we want to find the solution to $x - y > 5$ . We can start by graphing the related equation $x - y = 5$ .

When identifying the solution region, it is important not to assume that the solution will be above the line because of a “>” symbol or below the line because of a “<” symbol.

Instead, test the points on the line and on either side of the line, and see if they are solutions.

For $x-y>5$ , points on the line and above the line are not solutions to the inequality because the $(x,y)$ pairs make the inequality false. Points that are below the lines are solutions, so we can shade that lower region.

Graphing technology can help us graph the solution to an inequality in two variables.

Many graphing tools allow us to enter inequalities such as $x-y >5$ and will show the solution region, as shown here.

Some tools, however, may require the inequalities to be in slope-intercept form or another form before displaying the solution region. Be sure to learn how to use the graphing technology available in your classroom.

<p>Inequality graphed on a coordinate plane.</p>

Although graphing using technology is efficient, we still need to analyze the graph with care. Here are some things to consider:

The graphing window. If the graphing window is too small, we may not be able to really see the solution region or the boundary line, as shown here.
The meaning of solution points in the situation. For example, if $x$ and $y$ represent the lengths of two sides of a rectangle, then only positive values of $x$ and $y$ (or points in the first quadrant) make sense in the situation.

The Band Played On (1 problem)

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.

<p>Graph of inequality. Horizontal axis, x, from 0 to 275 by 25s. Vertical axis, y, from 0 to 275 by 25s. Line from 250 on vertical axis to 200 on horizontal axis. Shading above.</p>

Section B Check

Section B Checkpoint

Lesson 7

Solutions to Systems of Linear Inequalities in Two Variables

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In this lesson, two linear inequalities in two variables represent the constraints in a situation. Each pair of inequalities forms a system of inequalities.

A solution to a system of inequalities is any $(x,y)$ pair that makes both inequalities true, or any pair of values that simultaneously meet both constraints in the situation. The solution to the system is often best represented by a region on a graph.

Suppose there are two numbers, $x$ and $y$ , and there are two things we know about them.

The value of one number is more than double the value of the other.
The sum of the two numbers is less than 10.

We can represent these constraints with a system of inequalities.

$\begin {cases} y > 2x\\ x+y <10 \end {cases}$

There are many possible pairs of numbers that meet the first constraint, for example: 1 and 3, or 4 and 9.

The same can be said about the second constraint, for example: 1 and 3, or 2.4 and 7.5.

The pair $x=1$ and $y=3$ meets both constraints, so it is a solution to the system.

The pair $x=4$ and $y=9$ meets the first constraint but not the second ( $9 >2(4)$ is a true statement, but $4+9<10$ is not true.)

Remember that graphing is a great way to show all the possible solutions to an inequality, so let’s graph the solution region for each inequality.

<p>A graph of an inequality on a coordinate plane.</p>

Because we are looking for a pair of numbers that meet both constraints or make both inequalities true at the same time, we want to find points that are in the solution regions of both graphs.

To do that, we can graph both inequalities on the same coordinate plane.

The solution set to the system of inequalities is represented by the region where the two graphs overlap.

<p>A graph of two intersecting inequalities on a coordinate plane.</p>

Oh Good, Another Riddle (1 problem)

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 9

Modeling with Systems of Inequalities in Two Variables

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No cool-down

Section C Check

Section C Checkpoint

Unit 4 Linear Inequalities And Systems — Unit Plan