# Linear Equations, Inequalities and Systems

Students manipulate, graph, and model with two-variable linear equations and inequalities, are introduced to inverse functions, and continue studying linear systems of equations and inequalities.

## Unit Summary

In Unit 4, Linear Equations, Inequalities, and Systems, students become proficient at manipulating, identifying features, graphing, and modeling with two-variable linear equations and inequalities. Students are introduced to inverse functions and formalize their understanding on linear systems of equations and inequalities to model and analyze contextual situations. Proficiency of algebraic manipulation and solving, graphing skills, and identification of features of functions are essential groundwork to build future concepts studied in Units 5, 6, 7, and 8.

Topic A builds on work from Unit 3 to expand the idea of a solution to a coordinate point and to review identifying features of linear functions as well as graphing and writing equations in different forms to reveal properties. Students build on conceptual work from eighth grade on independence and dependence to define, create, and model with inverse functions.

Topic B expands students’ understanding of a single-variable inequality to linear inequalities. Students are expected to use tools of checking solutions strategically as well as attending to precision in notation and graphing.

Topic C combines learning from topics A and B to explore and model with systems of equations and inequalities. Students need to be precise in their calculations and choose efficient methods of solving as well as contextualize and decontextualize situations that can be modeled with a system of equations or inequalities. The unit concludes with a two-day, teacher-designed project.

## Assessment

This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Intellectual Prep

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Internalization of Standards via the Unit Assessment

• Take unit assessment. Annotate for:
• Standards that each question aligns to
• Purpose of each question: spiral, foundational, mastery, developing
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that assessment points to

Internalization of Trajectory of Unit

• Read and annotate "Unit Summary."
• Notice the progression of concepts through the unit using "Unit at a Glance."
• Essential understandings
• Connection to assessment questions

### Essential Understandings

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• Linear equations and linear inequalities can be used to model situations. These models can be used to describe the situation, to provide a generalization, and as a prediction tool by defining variables and representing the solution in the context of the problem.
• Linear equations and linear inequalities can be represented in graphs, multiple forms of equations, tables, and contextual situations—each highlighting particular features of the linear equation or inequality. Using all of these tools will help to make meaning of the situation that the inequality or equation models.
• A situation can be modeled by the intersection of two or more equations or inequalities called a system. Algebraic and graphical tools can be used to solve these systems.

### Vocabulary

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 Standard form Inverse functions ${f^{-1}( x)}$ Slope-intercept form & Point-slope form Linear inequality Linear equation Solution set Rate of change Boundary line Slope/y-intercept/zeros Systems of linear equations Constraint Systems of linear inequalities Solve by substitution/Solve by elimination Solution to a system

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### Building Functions
• F.BF.B.4 — Find inverse functions.

• F.BF.B.4.A — Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x—1) for x ? 1.

##### Creating Equations
• A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

• A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

##### High School — Number and Quantity
• N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.

##### Interpreting Functions
• F.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

• F.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

• F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

• F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

• F.IF.B.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

• F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

• F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.

• F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

##### Linear, Quadratic, and Exponential Models
• F.LE.A.1.A — Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

##### Reasoning with Equations and Inequalities
• A.REI.A.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.

• A.REI.C.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.

• A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

• A.REI.D.10 — Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

• A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

• A.REI.D.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.

##### Seeing Structure in Expressions
• A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

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• 7.EE.B.4.B

• 8.EE.B.5

• 8.EE.B.6

• 8.EE.C.7

• 8.EE.C.8

• 8.F.A.1

• 8.F.A.2

• 8.F.A.3

• 8.F.B.4

• 8.F.B.5

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• F.BF.A.1

• HSA-CED.A

• F.IF.B.4

• A.REI.A.2

• A.REI.C.7

• HSA-REI.D

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.