unit 1 equations and inequalities homework 4

• List of Lessons
• 1.0 Marking the Text in Mathematics
• 1.1 Order of Operations
• 1.2 Translating Verbal Phrases
• 1.3 Functions as Rules and Tables
• 1.4 Functions as Graphs
• Unit 1 Review
• Unit 1 Skillz Review Help
• 2.1 Real Numbers
• 2.2 Add and Subtract Real Numbers
• 2.3 Multiply and Divide Real Numbers
• 2.4 Combine Like Terms and Distributive Property
• Unit 2 Review
• Unit 2 Skillz Review
• 3.1 One Step Equations
• 3.2 Two Step Equations
• 3.3 MultiStep Equations
• 3.4 Variables on Both Sides of the Equation
• Unit 3 Review
• 4.1 Ratios and Proportions
• 4.2 Solving Proportions using Cross Products
• 4.3 Solving Percent Problems
• 4.4 Solving for Y
• Unit 4 Shortcuts
• Unit 4 Review
• 5.1 Plot Points in a Coordinate Plane
• 5.2 Graphing Linear Equations and Using Intercepts
• 5.3 Slope and Rate of Change
• 5.4 Graph in Slope-Intercept Form
• 5.5 Graph Linear Functions
• Unit 5 Review
• 6.1 Write Equations in Slope Intercept Form
• 6.2 Use Equations in Slope Intercept Form
• 6.3 Equations in Parallel/Perpendicular Form
• 6.4 Fit Line To Data
• Unit 6 Review
• Unit 6 Skillz Review Help
• SEMESTER EXAM REVIEW
• 7.1 Inequalities
• 7.2 Solving Inequalities
• 7.3 Multi-step Inequalities
• 7.4 Absolute Value Equations
• 7.5 Linear Inequalities in Two Variables
• Unit 7 Skillz Review Help
• 8.1 Solving Systems by Graphing
• 8.2 Solving Systems using Substitution
• 8.3 Solving Systems using Elimination
• 8.4 Solving Special Case Systems
• 8.5 Solving Systems of Linear Inequalities
• Unit 8 Review
• 9.1 Expand and Condense Exponents
• 9.2 Exponent Rules
• 9.3 Zero and Negative Exponents
• 9.4 Scientific Notation
• Unit 9 Review
• 10.1 Add and Subtract Polynomials
• 10.2 Multiply Polynomials
• 10.3 Polynomial Equations in Factored Form
• 10.4 Factoring Trinomials
• 10.5 Solving Quadratics by Factoring
• 10.6 Double Factoring
• Unit 10 Review
• Unit 10 Skillz Review Help
• 11.1 Simplifying Radicals
• 11.2 Operations with Square Roots
• Unit 11 Review
• 12.1 Graphing Quadratics
• 12.2 Solve Quadratics by Graphing
• 12.3 Solve Quadratics Using Square Roots
• 12.4 Solve Quadratics Using the Quadratic Formula
• Unit 12 Review
• Unit 12 Skillz Review
• SEMESTER 2 EXAM REVIEW
• Teacher Resources
• Flippedmath.com

• Precalculus
• Signup / Login

Unit 4: Systems of Linear Equations and Inequalities

Unit 1: generalizing patterns, day 1: intro to unit 1, day 2: equations that describe patterns, day 3: describing arithmetic patterns, day 4: making use of structure, day 5: review 1.1-1.3, day 6: quiz 1.1 to 1.3, day 7: writing explicit rules for patterns, day 8: patterns and equivalent expressions, day 9: describing geometric patterns, day 10: connecting patterns across multiple representations, day 11: review 1.4-1.7, day 12: quiz 1.4 to 1.7, day 13: unit 1 review, day 14: unit 1 test, unit 2: linear relationships, day 1: proportional reasoning, day 2: proportional relationships in the coordinate plane, day 3: slope of a line, day 4: linear equations, day 5: review 2.1-2.4, day 6: quiz 2.1 to 2.4, day 7: graphing lines, day 8: linear reasoning, day 9: horizontal and vertical lines, day 10: standard form of a line, day 11: review 2.5-2.8, day 12: quiz 2.5 to 2.8, day 13: unit 2 review, day 14: unit 2 test, unit 3: solving linear equations and inequalities, day 1: intro to unit 3, day 2: exploring equivalence, day 3: representing and solving linear problems, day 4: solving linear equations by balancing, day 5: reasoning with linear equations, day 6: solving equations using inverse operations, day 7: review 3.1-3.5, day 8: quiz 3.1 to 3.5, day 9: representing scenarios with inequalities, day 10: solutions to 1-variable inequalities, day 11: reasoning with inequalities, day 12: writing and solving inequalities, day 13: review 3.6-3.9, day 14: quiz 3.6 to 3.9, day 15: unit 3 review, day 16: unit 3 test, day 1: intro to unit 4, day 2: interpreting linear systems in context, day 3: interpreting solutions to a linear system graphically, day 4: substitution, day 5: review 4.1- 4.3, day 6: quiz 4.1 to 4.3, day 7: solving linear systems using elimination, day 8: determining number of solutions algebraically, day 9: graphing linear inequalities in two variables, day 10: writing and solving systems of linear inequalities, day 11: review 4.4- 4.7, day 12: quiz 4.4 to 4.7, day 13: unit 4 review, day 14: unit 4 test, unit 5: functions, day 1: using and interpreting function notation, day 2: concept of a function, day 3: functions in multiple representations, day 4: interpreting graphs of functions, day 5: review 5.1-5.4, day 6: quiz 5.1 to 5.4, day 7: from sequences to functions, day 8: linear functions, day 9: piecewise functions, day 10: average rate of change, day 11: review 5.5-5.8, day 12: quiz 5.5 to 5.8, day 13: unit 5 review, day 14: unit 5 test, unit 6: working with nonlinear functions, day 1: nonlinear growth, day 2: step functions, day 3: absolute value functions, day 4: solving an absolute value function, day 5: review 6.1-6.4, day 6: quiz 6.1 to 6.4, day 7: exponent rules, day 8: power functions, day 9: square root and root functions, day 10: radicals and rational exponents, day 11: solving equations, day 12: review 6.5-6.9, day 13: quiz 6.5 to 6.9, day 14: unit 6 review, day 15: unit 6 test, unit 7: quadratic functions, day 1: quadratic growth, day 2: the parent function, day 3: transforming quadratic functions, day 4: features of quadratic functions, day 5: forms of quadratic functions, day 6: review 7.1-7.5, day 7: quiz 7.1 to 7.5, day 8: writing quadratics in factored form, day 9: solving quadratics using the zero product property, day 10: solving quadratics using symmetry, day 11: review 7.6-7.8, day 12: quiz 7.6 to 7.8, day 13: quadratic models, day 14: unit 7 review, day 15: unit 7 test, unit 8: exponential functions, day 1: geometric sequences: from recursive to explicit, day 2: exponential functions, day 3: graphs of the parent exponential functions, day 4: transformations of exponential functions, day 5: review 8.1-8.4, day 6: quiz 8.1 to 8.4, day 7: working with exponential functions, day 8: interpreting models for exponential growth and decay, day 9: constructing exponential models, day 10: rational exponents in context, day 11: review 8.5-8.8, day 12: quiz 8.5 to 8.8, day 13: unit 8 review, day 14: unit 8 test, day 1:   intro to unit 4, day 2:   interpreting linear systems in context, day 3:   interpreting solutions to a linear system graphically, day 4:   substitution, day 5:   review 4.1- 4.3, day 6:   quiz 4.1 to 4.3, day 7:   solving linear systems using elimination, day 8:   determining number of solutions algebraically, day 9:   graphing linear inequalities in two variables, day 10:   writing and solving systems of linear inequalities, day 11:   review 4.4- 4.7, day 12:   quiz 4.4 to 4.7, day 13:   unit 4 review, day 14:   unit 4 test, math medic help.

Math  /  9th Grade  /  Unit 4: Linear Equations, Inequalities and Systems

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, 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. 

Pacing: 17 instructional days (14 lessons, 2 flex days, 1 assessment day)

Fishtank Plus for Math

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

The following assessments accompany Unit 4.

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Post-Unit Assessment Answer Key

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

• 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

The central mathematical concepts that students will come to understand in this unit

• 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. 

Terms and notation that students learn or use in the unit

Topic A: Properties and Solutions of Two-Variable Linear Equations and Inverse Functions

Identify the solutions and features of a linear equation and when two linear equations have the same solutions.

A.REI.D.10 A.SSE.B.3

Write linear equations given features, points, or graph in standard form, point-slope form, and slope-intercept form.

A.SSE.B.3 F.IF.B.4 F.IF.C.7.A

Determine if a function is linear based on the rate of change of points in the function presented graphically and in a table of values.

F.IF.B.6 F.IF.C.7.A F.IF.C.9 F.LE.A.1.A

Identify inverse functions graphically and from a table of values in contextual and non-contextual situations.

F.BF.B.4.A F.IF.A.1 F.IF.A.2 F.IF.B.5

Find inverse functions algebraically, and model inverse functions from contextual situations.

A.CED.A.4 F.BF.B.4.A F.IF.B.6

Create a free account to access thousands of lesson plans.

Already have an account? Sign In

Topic B: Properties and Solutions of Two-Variable Linear Inequalities

Describe the solutions and features of a linear inequality. Graph linear inequalities.

Write linear inequalities from graphs.

A.CED.A.3 A.REI.D.12

Write linear inequalities from contextual situations.

Topic C: Systems of Equations and Inequalities

Solve a system of linear equations graphically.

A.CED.A.3 A.REI.D.11

Identify solutions to systems of inequalities graphically. Write systems of inequalities from graphs and word problems.

Solve linear systems of equations of two variables by substitution.

A.CED.A.3 A.REI.C.5 A.REI.C.6 N.Q.A.2

Identify solutions to systems of equations algebraically using elimination. Write systems of equations.

Identify solutions to systems of equations using any method. Write systems of equations.

A.REI.A.1 A.REI.C.6 A.SSE.B.3

Identify solutions to systems of equations with three variables.

Common Core Standards

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

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 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

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.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Expressions and Equations

7.EE.B.4.B — Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

8.EE.B.5 — Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

8.EE.C.7 — Solve linear equations in one variable.

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

8.F.A.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.

8.F.A.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.B.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Future Standards

Standards in future grades or units that connect to the content in this unit

F.BF.A.1 — Write a function that describes a relationship between two quantities 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.

HSA-CED.A — Create equations that describe numbers or relationships

A.REI.A.2 — Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A.REI.C.7 — Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.

HSA-REI.D — Represent and solve equations and inequalities graphically

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.

Linear Expressions & Single-Variable Equations/Inequalities

Functions and Transformations

Request a Demo

See all of the features of Fishtank in action and begin the conversation about adoption.

Learn more about Fishtank Learning School Adoption.

Contact Information

School information, what courses are you interested in, are you interested in onboarding professional learning for your teachers and instructional leaders, any other information you would like to provide about your school.

Effective Instruction Made Easy

Access rigorous, relevant, and adaptable math lesson plans for free

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Chapter 4: Inequalities

4.1 Solve and Graph Linear Inequalities

When given an equation, such as [latex]x = 4[/latex] or [latex]x = -5,[/latex] there are specific values for the variable. However, with inequalities, there is a range of values for the variable rather than a defined value. To write the inequality, use the following notation and symbols:

Example 4.1.1

Given a variable [latex]x[/latex] such that [latex]x[/latex] > [latex]4[/latex], this means that [latex]x[/latex] can be as close to 4 as possible but always larger. For [latex]x[/latex] > [latex]4[/latex], [latex]x[/latex] can equal 5, 6, 7, 199. Even [latex]x =[/latex] 4.000000000000001 is true, since [latex]x[/latex] is larger than 4, so all of these are solutions to the inequality. The line graph of this inequality is shown below:

Written in interval notation, [latex]x[/latex] > [latex]4[/latex] is shown as [latex](4, \infty)[/latex].

Example 4.1.2

Example 4.1.3

For greater than or equal (≥) and less than or equal (≤), the inequality starts at a defined number and then grows larger or smaller. For [latex]x \ge 4,[/latex] [latex]x[/latex] can equal 5, 6, 7, 199, or 4. The line graph of this inequality is shown below:

Written in interval notation, [latex]x \ge 4[/latex] is shown as [latex][4, \infty)[/latex].

Example 4.1.4

If [latex]x \le 3[/latex], then [latex]x[/latex] can be any value less than or equal to 3, such as 2, 1, −102, or 3. The line graph of this inequality is shown below:

Written in interval notation, [latex]x \le 3[/latex] is shown as [latex](-\infty, 3].[/latex]

When solving inequalities, the direction of the inequality sign (called the sense) can flip over. The sense will flip under two conditions:

Example 4.1.5

Solve the inequality [latex]5-2x[/latex] > [latex]11[/latex] and show the solution on both a number line and in interval notation.

First, subtract 5 from both sides:

[latex]\begin{array}{rrrrr} 5&-&2x&\ge &11 \\ -5&&&&-5 \\ \hline &&-2x&\ge &6 \end{array}[/latex]

Divide both sides by −2:

[latex]\begin{array}{rrr} \dfrac{-2x}{-2} &\ge &\dfrac{6}{-2} \\ \end{array}[/latex]

Since the inequality is divided by a negative, it is necessary to flip the direction of the sense.

This leaves:

[latex]x \le -3[/latex]

In interval notation, the solution is written as [latex](-\infty, -3][/latex].

On a number line, the solution looks like:

Inequalities can get as complex as the linear equations previously solved in this textbook. All the same patterns for solving inequalities are used for solving linear equations.

Example 4.1.6

The solution written on a number line is:

For questions 1 to 6, draw a graph for each inequality and give its interval notation.

• [latex]n[/latex] > [latex]-5[/latex]
• [latex]n[/latex] > [latex]4[/latex]
• [latex]-2 \le k[/latex]
• [latex]1 \ge k[/latex]
• [latex]5 \ge x[/latex]
• [latex]-5 < x[/latex]

For questions 7 to 12, write the inequality represented on each number line and give its interval notation.

For questions 13 to 38, draw a graph for each inequality and give its interval notation.

• [latex]\dfrac{x}{11}\ge 10[/latex]
• [latex]-2 \le \dfrac{n}{13}[/latex]
• [latex]2 + r < 3[/latex]
• [latex]\dfrac{m}{5} \le -\dfrac{6}{5}[/latex]
• [latex]8+\dfrac{n}{3}\ge 6[/latex]
• [latex]11[/latex] > [latex]8+\dfrac{x}{2}[/latex]
• [latex]2[/latex] > [latex]\dfrac{(a-2)}{5}[/latex]
• [latex]\dfrac{(v-9)}{-4} \le 2[/latex]
• [latex]-47 \ge 8 -5x[/latex]
• [latex]\dfrac{(6+x)}{12} \le -1[/latex]
• [latex]-2(3+k) < -44[/latex]
• [latex]-7n-10 \ge 60[/latex]
• [latex]18 < -2(-8+p)[/latex]
• [latex]5 \ge \dfrac{x}{5} + 1[/latex]
• [latex]24 \ge -6(m - 6)[/latex]
• [latex]-8(n - 5) \ge 0[/latex]
• [latex]-r -5(r - 6) < -18[/latex]
• [latex]-60 \ge -4( -6x - 3)[/latex]
• [latex]24 + 4b < 4(1 + 6b)[/latex]
• [latex]-8(2 - 2n) \ge -16 + n[/latex]
• [latex]-5v - 5 < -5(4v + 1)[/latex]
• [latex]-36 + 6x[/latex] > [latex]-8(x + 2) + 4x[/latex]
• [latex]4 + 2(a + 5) < -2( -a - 4)[/latex]
• [latex]3(n + 3) + 7(8 - 8n) < 5n + 5 + 2[/latex]
• [latex]-(k - 2)[/latex] > [latex]-k - 20[/latex]
• [latex]-(4 - 5p) + 3 \ge -2(8 - 5p)[/latex]

Answer Key 4.1

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.

Equations and Inequalities

Lesson 1 cafeteria actions and reactions, lesson 2 elvira’s equations, lesson 3 solving equations literally, lesson 4 greater than, lesson 5 may i have more, please, lesson 6 taking sides.

• Teacher Info
• Graphing Calculator

Math 1 Unit 4 Equations and inequalities

Unit 4 important dates, assignments.

• Genie help with Inequalities (Practice with Inequalities 1)
• Rags to Riches Quia Game ( Practice with Inequalities 2​ )
• BrainPop Practice Quiz (Practice with Inequalities 3)

​ tutorial links by topic

• Solving Equations with Variables on Both Sides
• Solving Equations with Variables on both sides - PRACTICE
• Solving Equations with Variables on both sides - ANOTHER PRACTICE
• Solving Equations with more than one Variable
• Solving Formulas (with more than one Variable)
• Solving Equations with more than one variable - PRACTICE
• Solving Inequalities PRACTICE  (with videos as well on the left tab)
• Solving Inequalities - Word Problem s