Coordinate Algebra: Unit 2 – Reasoning with Equations and Inequalities
(5 weeks)
Unit Overview: By the end of eighth grade, students have learned to solve linear equations in one variable and have applied
graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier
experiences by asking students to analyze and explain the process of solving an equation and to justify the process used in solving a
system of equations. Students develop fluency writing, interpreting, and translating between various forms of linear equations and
inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques
and the laws of exponents to the creation and solution of simple exponential equations. Students explore systems of equations and
inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on
relationships between them.
Content Standards:
Understand solving equations as a process of reasoning and explain the reasoning
MCC9-12.A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous
step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution
method. (Students should focus on and master linear equations and be able to extend and apply their reasoning to other types of
equations in future courses.)
Solve equations and inequalities in one variable
MCC9-12.A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by
letters. (Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations
that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of
exponents, such as 5x = 125 or 2x = 1/16.)
Solve systems of equations
MCC9-12.A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation
and a multiple of the other produces a system with the same solutions. (Limit to linear systems.)
MCC9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear
equations in two variables.
Represent and solve equations and inequalities graphically
MCC9-12.A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a
strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the
corresponding half-planes.
Standards for Mathematical Practice:
SMP3 Construct viable arguments and critique the reasoning of others.
SMP5 Use appropriate tools strategically.
SMP1 Make sense of problems and persevere in solving them.
SMP6 Attend to precision.
Diagnostic: Prerequisite Skills Assessment
Standards for Mathematical Practice (3, 7)
EQ: How do you know if you have a convincing argument? (MP3) What strategies do mathematically proficient students use to
identify patterns? (MP7)
Learning Targets:
I can …
 use definitions and previously established causes/effects (results) in constructing arguments (MP3)
 make conjectures and use counterexamples to build a logical progression of statements to explore and support their ideas
(MP3)
 communicate and defend mathematical reasoning using objects, drawings, and diagrams (MP3)
 listen to or read the arguments of others (MP3)
 decide if the arguments of others make sense and ask probing questions to clarify or improve the arguments (MP3)
 look closely to discern a pattern or structure. (MP7)
Concept Overview:
MP3 Construct viable arguments and critique the reasoning of others.
In this unit, students expand the previously learned concepts of solving and graphing linear equations and inequalities, focusing on
the reasoning and understanding involved in justifying the solution. Students are asked to explain and justify the mathematics
required to solve both simple equations and systems of equations in two variables using both graphing and algebraic methods.
Students explore systems of equations and inequalities, and provide justifications for why their methods work.
MP7 Look for and make use of structure.
Teachers who are developing students’ capacity to "look for and make use of structure" help learners identify and evaluate efficient
strategies for finding solutions. One instructional strategy that can develop this habit of mind is “I Notice/I Wonder.” Proficient
notices the structure that’s there; proficient wonders seek out even hidden structures. As students practice noticing and wondering
in the context of solving problems and learning content, they become better able to see and more disposed to look for structure.
The second is Solve a Simpler Problem. The activities there, while focused on standards methods of simplifying problems or looking
for analogous problems, exploit the structure of problems or of the objects within problems. When students look to see how they
could make a problem simpler and whether they’ve changed the problem, they are grappling with its underlying structure. When
they change a problem to an analogous problem or relate it to a problem they’ve solved before, they’re again using the ability to
see structure. And when they re-represent the numbers in the problem, they are realizing that numbers are flexible and they have
the power to decompose and recompose them as needed.
Resources:
MP3 Inside Mathematics Website
MP7 Inside Mathematics Website
Understanding Solving Equations as a Process of Reasoning
EQ: In what ways can problems be solved, and why should one method be chosen over another?
Learning Targets:
I can …
 Understand, apply, and explain the results of using inverse operations.
 Justify the steps in solving equations by applying and explaining the properties of equality, inverse and identity.
 Use the names of the properties and common sense explanations to explain the steps in solving an equation.
Concept Overview: Properties of operations can be used to change expressions on either side of the equation to equivalent
expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant
produces an equation with the same solutions. Other operations, such as squaring both sides, may produce equations that have
extraneous solutions. Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their
reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in CCGPS
Advanced Algebra.
Assuming an equation has a solution, have students construct a convincing argument that justifies each step in the solution process.
Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc.
Challenge students to justify each step of solving an equation. Transforming 2x – 5 = 7 to 2x = 12 is possible because 5 = 5, so adding
the same quantity to both sides of an equation makes the resulting equation true as well. Each step of solving an equation can be
defended, much like providing evidence for steps of a geometric proof.
Provide examples for how the same equation might be solved in a variety of ways as long as equivalent quantities are added or
subtracted to both sides of the equation, the order of steps taken will not matter.
3n +2 = n – 10
3n + 2 = n – 10
3n +2 = n – 10
-2 =
-2
+10 = +10
-n = -n__
3n = n – 12
or
or
3n +12 = n
2n +2 = -10
-n = -n__
-3n
= -3n__
-2 = -2__
2n = -12
12 = -2n
2n = -12
2
2
-2 -2
2
2
n = -6
n = -6
n = -6
The Properties of Operations
Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number
system, the real number system, and the complex number system.
Associative property of addition
Commutative property of addition
Additive identity property of 0
Existence of additive inverses
Associative property of multiplication
Commutative property of multiplication
Multiplicative identity property of 1
Existence of multiplicative inverses
Distributive property of multiplication over addition
(a + b) + c = a + (b + c)
a+b=b+a
a+0=0+a =a
For every a there exists –a so that a + (-a) = (-a) + a = 0.
(a *b) * c = a *(b * c)
a*b=b*a
a*1=1*a=a
1
1
1
For every a ≠0 there exists so that a* = *a = 1.
a
a
a * (b + c) = a * b + a * c
The Properties of Equality
Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.
Reflexive property of equality
Symmetric property of equality
Transitive property of equality
Addition property of equality
Subtraction property of equality
a=a
If a = b, then b = a
If a = b and b = c, then a = c
If a = b, then a + c = b + c
If a = b, then a – c = b – c
a
Multiplication property of equality
Division property of equality
Substitution property of equality
If a = b, then a ∗ c = b ∗ c
If a = b and c ≠ 0, then a ÷ c = b ÷ c
If a = b, then b may be substituted for a in any
expression containing a.
Vocabulary:
Properties of operations and properties of equalities – refer to the table above
Coefficient - the number in front of the variable in a given expression
Like terms – terms that have the same variables and the same corresponding exponents
Variable – a quantity that changes or that can have different values; a symbol, usually a letter, that can stand for a variable quantity
Evaluate – to find the value of a mathematical expression
Justify - to prove or show that the answer is valid
Viable - capable of working successfully
Inverse operation – the operation that reverses the effect of another operation
Equivalent expressions - expressions that are the same, even though they may look a little different. If you substitute in the same
variable value into equivalent expressions, they will each give you the same value when you simplify.
Extraneous solutions - a solution of the simplified form of an equation that does not satisfy the original equation.
Sample Problem(s):
Which of the following equations have the same solution? Give reasons for your answer that does not depend
on solving the equations. REI.1 (REI.1 solution)
1.
2.
3.
4.
5.
6.
x+3 = 5x−4
x−3 = 5x+4
2x+8 = 5x−3
10x+6 = 2x−8
10x−8 = 2x+6
0.3+
x
10
1
= x - 0.4
2
When Sally picks any number between 1 and 20, doubles it, adds 6, divides by 2 and subtracts 3, she always gets the number she
started with. Why? Evaluate and use algebraic evidence to support your conclusion. REI.1 Solution
Explain why the equation x/2 + 7/3 = 5 has the same solutions as the equation 3x + 14 = 30. Does this mean that x/2 + 7/3 is equal to
3x + 14? REI.1 Solution
Standard
MCC912.A.REI.1
Topic
Understand
solving equations
as a process of
reasoning
Resources
Additional resources:
 Project: “Mathemagic”
Tricks (M, U)
 Prentice Hall Notetaking
Guide 4.3 A/S Equality
 Prentice Hall Notetaking
Guide 4.4 D/M Equality
 Prentice Hall Notetaking
Guide 4.5
 Prentice Hall Notetaking
Guide 4.6 Inverse
Operation
Teacher Notes
Student Misconceptions:
Students may believe that solving an equation such as 3x +1=
7 involves “only removing the 1,” failing to realize that the
equation 1 = 1 is being subtracted to produce the next step.
Probing questions:
1. How are zero pairs used to solve equations?
2. What does isolating a variable mean?
3. Explain why the multiplication property of equality and the
division property of equality can be considered the same
property?
4. What does it mean to undo an operation in an equation?
Differentiation Strategy:
1. Allow students to use algebra tiles to model using zero
pairs
Cooperative Learning Strategy Partner work Algebraic
Properties
Literacy Strategy: Vocabulary Mathematical Metaphors
Properties (K)
Solving Literal Equations and Inequalities in One Variable
Solve equations and inequalities in one variable
MCC9-12.A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by
letters. (Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal
equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of
the laws of exponents, such as 5x = 125 or 2x = 1/16.)
EQ: In what ways can the problem be solved, and why should one method be chosen over another?
Learning Targets:
I can …
 Write equations in equivalent forms to solve problems. (REI.3)
 Analyze and solve literal equations for a specified variable. (REI.3)
 Understand and apply the properties of Inequalities. (REI.3)
 Verify that a given number or variable is a solution to the equation or inequality. (REI.3)
 Interpret the solution of an inequality in real terms. (REI.3)
 Solve linear equations and inequalities with one variable, including absolute value equations and inequalities and
compound inequalities. (REI.3)
 Solve an equation for a specific variable and reason about the new equation (i.e. A=1/2 Bh: if I know that the area of a
triangle is 50 units, then 100/h = B, so as h increases, B decreases.) (REI.3)
 Solve simple exponential equations using laws of exponents. (REI.3)
 Graph the solution to linear inequalities in two variables. (REI.3)
 Graph the solution to systems of linear inequalities in two variables. (REI.3)
 Identify the solutions as a region of the plane. (REI.3)
Concept Overview: In grades 6-8, students solve and graph linear equations and inequalities. There are two major reasons for
discussing the topic of inequalities along with equations: First, there are analogies between solving equations and inequalities
that help students understand them both. Second, the applications that lead to equations almost always lead in the same way
to inequalities. Graphing experience with inequalities should be limited to graphing on a number line diagram. Despite this
work, some students will still need more practice to be proficient. It may be beneficial to remind students of the most common
solving techniques, such as converting fractions from one form to another, removing parentheses in the sentences, or
multiplying both sides of an equation or inequality by the common denominator of the fractions. Students must be aware of
what it means to check an inequality’s solution. The substitution of the end points of the solution set in the original inequality
should give equality regardless of the presence or the absence of an equal sign in the original sentence. The substitution of any
value from the rest of the solution set should give a correct inequality.
Careful selection of examples and exercises is needed to provide students with meaningful review and to introduce other
important concepts, such as the use of properties and applications of solving linear equations and inequalities. Stress the idea
that the application of properties is also appropriate when working with equations or inequalities that include more than one
variable, fractions and decimals. Regardless of the type of numbers or variables in the equation or inequality, students have to
examine the validity of each step in the solution process.
Solving equations for the specified letter with coefficients represented by letters (e.g., A = ½(B + b) when solving for b) is similar
to solving an equation with one variable. Provide students with an opportunity to abstract from particular numbers and apply
the same kind of manipulations to formulas as they did to equations. One of the purposes of doing abstraction is to learn how
to evaluate the formulas in easier ways and use the techniques across mathematics and science.
Draw students’ attention to equations containing variables with subscripts. The same variables with different subscripts (e.g., x1
and x2) should be viewed as different variables that cannot be combined as like terms. A variable with a variable subscript, such
as an, must be treated as a single variable – the nth term, where variables a and n have different meaning.
The graphing method can be the first step in solving systems of equations. A set of points representing solutions of each
equation is found by graphing these equations. Even though the graphing method is limited in finding exact solutions and often
yields approximate values, the use of it helps to discover whether solutions exist and, if so, how many are there. The next step
is to turn to algebraic methods, elimination or substitution, to allow students to find exact solutions. For any method, stress the
importance of having a well-organized format for writing solutions.
Vocabulary:
Inequality –any mathematical sentence that contains the symbols > (greater than), < (less than), ≤ (less than or equal to), or ≥
(greater than or equal to)
Property of InequalitiesHere a, b and c stand for arbitrary numbers in the rational or real number systems.
Exactly one of the following is true: a < b, a = b, a > b.
If a > b and b > c then a > c.
If a > b, then b < a.
If a > b, then –a < –b.
If a > b, then a ± c > b ± c.
If a > b and c > 0, then a × c > b × c.
If a > b and c < 0, then a × c < b × c.
If a > b and c > 0, then a ÷ c > b ÷ c.
If a > b and c < 0, then a ÷ c < b ÷ c.
Solution – any value for a variable that makes an equation true
Literal equations- an equation where variables represent known values
Equivalent equations- equivalent equations are the equations that have the same solution
Sample Problem(s):
The perimeter of a rectangle is given by P = 2W + 2L. Solve for W and restate in words the meaning of this new formula in terms
of the meaning of the other variables. REI.3 Solution
Standard
MCC912.A.REI.3
Topic
Solving
equations as a
means for
solving literal
equations and
inequalities
Resources
Textbook Section #’s:
 (Online Textbook Codes)
 Holt 2-8 Solving Two-Step
Equations
 Holt Algebra I –Section 2-1
 Holt Algebra I – Section 2B
 Holt Algebra I– Section 2-5
 Prentice Hall Course 3 -- Pages
29-30; Pages 35-36 and Pages
277-278
Additional resources:
 Variables on Both Sides (S)
 Multi-Step Equations (S, U)
 One-Step Equations (S)
 Two-Step Equations (S)
 Inequalities and Graphs
Teacher Notes
Student Misconceptions:
Some students may believe that for equations
containing fractions only on one side, it requires
“clearing fractions” (the use of multiplication) only
on that side of the equation. When addressing the
misconception, start by demonstrating the solution
methods for equations similar to
1
1
1
x + x + x + 46 = x, and stress that the
4
5
6
multiplication Property of Equality is applied to
both sides, by multiplying by 60, which is the LCD.
Some students may believe that subscripts can be
combined as b1 + b2 = b3 and the sum of different
variables d and D is d +D = 2D.
Some students may think that rewriting equations
into various forms (taking square roots, completing
the square, using quadratic formula and factoring)
are isolated techniques within a unit of quadratic
equations. Teachers should help students see the
value of these skills in the context of solving higher
degree equations and examining different families
of functions.
Probing questions:
1. Explain how an inequality with one variable
differs from an equation with one variable.
2. How many solutions can an equation have?
Differentiation Strategy:
1. Allow students to use the TRACE feature on the
graphing calculator to solve equations. Students
will see the solution is the x value that corresponds
to the same two y-values in the table.
2. Have students use algebra tiles to model solving
equations.
Cooperative Learning Strategy: Rally Robin
Describe the steps needed to solve this equation or
inequality.
Literacy Strategy: Alike and Different Equations and
Inequalities (K)
MCC912.A.REI.3
Solving
inequalities
Textbook Section #’s:
(Online Textbook Codes)
 Pearson Course 3 p. 281-285
 Pearson Course 3 6-5 Resources
 Pearson Course 3 pp. 288-291
 Pearson Course 3 6-6 Resources
 Holt 3-4 and 3-5
 Holt 6-5
Additional resources:
 Multi-Step Inequalities (U)
 One-Step Inequalities (K)
 Jaden’s Phone Plan
(from state frameworks)
 Absolute Value Inequalities
 Compound Inequalities
Unit 2 Frameworks TE with answers
Model Lesson: (S) Students will be introduced to
inequalities in two variables and will see that the
solution set consists of all points in a half-plane.
Student Misconceptions:
Students may confuse the rule of changing a sign of
an inequality when multiplying or dividing by a
negative number with changing the sign of an
inequality when one or two sides of the inequality
become negative (for ex., 3x > -15 or x < - 5).
Probing questions:
1. Explain how an inequality with one variable
differs from an equation with one variable.
2. Why is graphing the solutions of an inequality
more efficient than listing all the solutions of the
inequality.
3. How many solutions can an inequality have?
4. What must you remember to do to an inequality
when dividing by a negative number?
Differentiation Strategy:
1. Have students circle the inequality when dividing
by a negative number to help them remember to
reverse the inequality symbol)
Cooperative Learning Strategy: Four Corners
Solution to Inequalities
Verifying Solutions to Systems of Equations
Solve systems of equations
MCC9-12.A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation
and a multiple of the other produces a system with the same solutions. (Limit to linear systems.)
MCC9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear
equations in two variables.
EQ: How can a solution for a given System of Equations be verified?
Learning Targets:
I can …
 Explain the use of the multiplication property of equality to solve a system of equations. (REI.5)
 Explain why the sum of two equations is justifiable in the solving of a system of equations (property of equality). (REI.5)
 Relate the process of linear combinations with the process of substitution for solving a system of linear equations. (REI.5)
 Explain how to solve a system of two equations and justify the method used to find the solution. (REI.5)
 Explain why replacing one equation by the sum of that equation with a multiple of the other produces a system with the
same solution. (REI.6)
 Create a system of equations to represent relationships between quantities. (REI.6)
Concept Overview: The focus of this standard is to provide mathematics justification for the addition (elimination) and substitution
methods of solving systems of equations that transform a given system of two equations into a simpler equivalent system that has
the same solutions as the original.
The Addition and Multiplication Properties of Equality allow finding solutions to certain systems of equations. In general, any linear
combination, m(Ax + By) + n(Cx + Dy) = mE +nF, of two linear equations
Ax + By = E and
Cx + Dy = F
intersecting in a single point contains that point. The multipliers m and n can be chosen so that the resulting combination has only
an x-term or only a y-term in it. That is, the combination will be a horizontal or vertical line containing the point of intersection.
In the proof of a system of two equations in two variables, where one equation is replaced by the sum of that equation and a
multiple of the other equation, produces a system that has the same solutions, let point (x1, y1) be a solution of both equations:
Ax1 + By1 = E (true)
Cx1 + Dy1 = F (true)
Replace the equation Ax + By = E with Ax + By + k(Cx + Dy) on its left side and with E + kF on its right side.
The new equation is Ax + By + k(Cx + Dy) = E + kF.
Show that the ordered pair of numbers (x1, y1) is a solution of this equation by replacing (x1, y1) in the left side of this equation and
verifying that the right side really equals E + kF:
Ax1 + By1 + k(Cx1 + Dy1) = E + kF (true)
Systems of equations are classified into two groups, consistent or inconsistent, depending on whether or not solutions exist. The
solution set of a system of equations is the intersection of the solution sets for the individual equations. Stress the benefit of making
the appropriate selection of a method for solving systems (graphing vs. addition vs. substitution). This depends on the type of
equations and combination of coefficients for corresponding variables, without giving a preference to either method.
Begin with simple, real-world examples and help students to recognize a graph as a set of solutions to an equation. For example, if
the equation y = 6x + 5 represents the amount of money paid to a babysitter (i.e., $5 for gas to drive to the job and $6/hour to do
the work), then every point on the line represents an amount of money paid, given the amount of time worked.
Explore visual ways to solve an equation such as 2x + 3 = x – 7 by graphing the functions y = 2x + 3 and y = x – 7. Students should
recognize that the intersection point of the lines is at (-10, -17). They should be able to verbalize that the intersection point means
that when x = -10 is substituted into both sides of the equation, each side simplifies to a value of -17. Therefore, -10 is the solution
to the equation. This same approach can be used whether the functions in the original equation are linear, nonlinear or both.
Using technology, have students graph a function and use the trace function to move the cursor along the curve. Discuss the
meaning of the ordered pairs that appear at the bottom of the calculator, emphasizing that every point on the curve represents a
solution to the equation.
Begin with simple linear equations and how to solve them using the graphs and tables on a graphing calculator. Then, advance
students to nonlinear situations so they can see that even complex equations that might involve quadratics, absolute value, or
rational functions can be solved fairly easily using this same strategy. While a standard graphing calculator does not simply solve an
equation for the user, it can be used as a tool to approximate solutions.
Use the table function on a graphing calculator to solve equations. For example, to solve the equation x2 = x + 12, students can
examine the equations y = x2 and y = x + 12 and determine that they intersect when x = 4 and when x = -3 by examining the table to
find where the y-values are the same.
Vocabulary:
System of equations- is a collection of one or more linear equations, with the same set of unknowns.
Solution set - the intersection of the solution sets for the individual equations
Elimination method- is the process of eliminating one of the variables in a system of equations using addition or subtraction in
conjunction with multiplication or division and solving the system of equations
Elimination by multiplication – multiplying and combining equations in a system in order to eliminate a variable.
Elimination by addition – adding and combining equations in a system in order to eliminate a variable
Elimination by substitution – solving one of the equations for one variable in terms of the other, and then substitute that into the
other equation.
Sample Problem(s):
Create a new system using both the addition and multiplication properties of equality. Then verify that the new system has the
same solution as the original. REI.5 Solution
2𝑥 + 𝑦 = 5
{
}
−5𝑥 − 2𝑦 = −6
Standard
MCC912.A.REI.5
Topic
Equivalent systems
Resources
Textbook Section #’s:
(Online Textbook Codes)
 Holt Section 6-3
Additional Resources:
 Systems of Equations W1
 Systems of Equations W2
Teacher Notes
Model Lesson (U)
Student Misconceptions: Most mistakes that students
make are carless rather than conceptual. Teachers
should encourage students to learn a certain format for
solving systems of equations and check the answers by
substituting into all equations in the system.
Probing questions:
1. What relationship exists between two dependent
systems?
Differentiation Strategy:
1. Have students solve each equation for y or get the
equation in slope-intercept form. They can use the slope
to determine if the two lines are the same (consistent)
or parallel (inconsistent).
Cooperative Learning Strategy: Mix ‘n’ Match
Equivalent Systems
Literacy Strategy: Vocabulary Words with Multiple
Meanings (K)
Verifying Solutions to Systems of Inequalities
Represent and solve equations and inequalities graphically
MCC9-12.A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a
strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the
corresponding half-planes.
Represent and solve equations and inequalities graphically
MCC9-12.A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a
strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the
corresponding half-planes.
EQ: How can a solution for a given System of Equations be verified?
Learning Targets:
I can…
 Explain why replacing one equation by the sum of that equation with a multiple of the other produces a system with the same
solution. (REI.6)
 Create a system of equations to represent relationships between quantities. (REI.6)
 Graph a system of linear inequalities on a coordinate plane. (REI.12)
 Explain that the solution set for a system of linear inequalities is the intersection of the shaded regions (half-planes) of both
inequalities. (REI.12)
 Check points in the intersection of the half-planes to verify that they represent a solution to the system. (REI.12)
Concept Overview:
Investigate real-world examples of two-dimensional inequalities. For example, students might explore what the graph would look
like for money earned when a person earns at least $6 per hour. (The graph for a person earning exactly $6/hour would be a linear
function, while the graph for a person earning at least $6/hour would be a half-plane including the line and all points above it.)
Applications such as linear programming can help students to recognize how businesses use constraints to maximize profit while
minimizing the use of resources. These situations often involve the use of systems of two variable inequalities.
Vocabulary:
Half-plane – the result of graphing an inequality on the coordinate plane, which creates a boundary that cuts the coordinate plane
in half.
System of linear inequalities- set of two or more linear inequalities containing two or more variables
Solution of system of linear inequalities- all the ordered pairs that satisfy all the linear inequalities in the system.
Solution region – all points (a, b) such that when x = a, and y = b, all inequalities are true.
Sample Problem(s):
A publishing company publishes a total of no more than 100 magazines every year. At least 30 of these are women’s magazines, but
the company always publishes at least as many women’s magazines as men’s magazines. Find a system of inequalities that describes
the possible number of men’s and women’s magazines that the company can produce each year consistent with these policies.
Graph the solution set. REI.6 Solution
Create a context that represents the shaded area. Write the system of inequalities that models the meaning of the context. Describe
the connections between the contexts, inequalities, and graphs.
Standard
Topic
Resources
Textbook Section #’s:
Teacher Notes
Model Lesson: (M, S) Students will use graphical
MCC912.A.REI.6
Solving systems
of equations
exactly and
approximately
(Online Textbook Codes)
 Holt Section 6-1
 Holt Section 6-2
 Holt Section 6-3
 Holt Section 6-4
 Holt Section 6-5
 Holt Section 6-6
and algebraic techniques to solve an application
problem involving systems of equations.
Additional resources:
 Graphic Organizer: Methods of
Solving Systems of Equations (S)
 Solving Systems by Graphing 1 (K)
 Solving Systems by Graphing 2 (K)
 Solving Systems by Substitution (K)
 Systems of Equations Word
Problems (M)
 Systems Word Problems (M)
 Solving Systems Algebraically
(from state frameworks)
Additionally, students may believe that systems of
inequalities have no application in the real world.
Teachers can consider business related problems
(e.g., linear programming applications) to engage
students in discussions of how the systems
inequalities are derived and how the feasible set
includes all the points that satisfy the conditions
stated in the inequalities.
Student Misconceptions: Students may believe that
graphing linear and other functions is an isolated
skill, not realizing that multiple graphs can be drawn
to solve equations involving those functions.
Probing questions:
1. How can you find exact solutions to a system of
equations without using a graphing calculator?
2. How can you use multiplication to create a system
of equations that are equivalent?
3. Does solving a system algebraically always result
in a unique system? Why?
4. How do you decide whether to solve a system
using elimination by substitution, elimination by
multiplication, and elimination by addition?
Differentiation Strategy:
1. Have students solve each equation for y or get
the equation in slope-intercept form. They can use
the slope to determine if the two lines are the same
(consistent) or parallel (inconsistent).)
Cooperative Learning Strategy: Group Activity
Battleship
Literacy Strategy: Creating an Outline to Solidify
Meaning (K)
MCC912.A.REI.12
Solving Systems
of inequalities
Additional resources:
 Systems of Linear Inequalities (S)
 Project: Making Cookies, Making
Money (Linear Programming) (M)
 Notetaking Guide Graphing
Inequalities
 Summer Job
(from state frameworks)
 Graphing Inequalities
(from state frameworks)
Model Lesson: (M, S) Students will use graphical
and algebraic techniques to solve an application
problem involving systems of equations.
Student Misconceptions: Students may believe that
systems of inequalities have no application in the
real world. Teachers can consider business related
problems (e.g., linear programming applications) to
engage students in discussions of how the systems
inequalities are derived and how the feasible set
includes all the points that satisfy the conditions
stated in the inequalities.
The solution to the system will be a whole number,
many application problems may be restricted to
whole numbers, but not all solutions will be whole
numbers.
Probing questions:
1. Describe the two half-planes created by the
boundary line created by the graph of an inequality?
2. How will you decide which side of the boundary
line or which half-plane to shade?
3. What is the difference between a half-plane for a
graph of a greater than ( >) or less than (< )
inequality and a greater than or equal to (≥) or less
than or equal to (≤ ) inequality?
Differentiation Strategy:
1. Use the SHADE feature of a graphing calculator to
help students graph inequalities.
2. Have students solve for y on the left side of the
inequality to help decide whether to include the
boundary in the graph and what region of the halfplane to shade.
Cooperative Learning Strategy: Timed Pair Share
Solving Systems of Inequalities
Literacy Strategy: Creating an Outline to Solidify
Meaning (K)
Unit #2 Summative Assessment