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 7: Inequalities (systems & graphs)
About this unit.
If it takes someone in a cooking competition 30 seconds to chop an onion and 18 seconds to chop a carrot, and their goal is to chop at least 20 of these vegetables in a time limit of 540 seconds, what can they do? Use a system of linear inequalities, of course.
Checking solutions of two-variable inequalities
- Testing solutions to inequalities (Opens a modal)
- Testing solutions to systems of inequalities (Opens a modal)
- Solutions of inequalities: algebraic Get 3 of 4 questions to level up!
- Solutions of inequalities: graphical Get 3 of 4 questions to level up!
- Solutions of systems of inequalities Get 3 of 4 questions to level up!
Graphing two-variable inequalities
- Intro to graphing two-variable inequalities (Opens a modal)
- Graphing two-variable inequalities (Opens a modal)
- Two-variable inequalities from their graphs (Opens a modal)
- Intro to graphing systems of inequalities (Opens a modal)
- Graphing systems of inequalities (Opens a modal)
- Graphing inequalities (x-y plane) review (Opens a modal)
- Graphs of inequalities Get 3 of 4 questions to level up!
- Two-variable inequalities from their graphs Get 3 of 4 questions to level up!
- Systems of inequalities graphs Get 3 of 4 questions to level up!
Modeling with linear inequalities
- Writing two-variable inequalities word problem (Opens a modal)
- Solving two-variable inequalities word problem (Opens a modal)
- Interpreting two-variable inequalities word problem (Opens a modal)
- Modeling with systems of inequalities (Opens a modal)
- Writing systems of inequalities word problem (Opens a modal)
- Solving systems of inequalities word problem (Opens a modal)
- Graphs of systems of inequalities word problem (Opens a modal)
- Graphs of two-variable inequalities word problem (Opens a modal)
- Inequalities (systems & graphs): FAQ (Opens a modal)
- Creativity break: What can we do to expand our creative skills? (Opens a modal)
- Two-variable inequalities word problems Get 3 of 4 questions to level up!
- Systems of inequalities word problems Get 3 of 4 questions to level up!
Common Core Mapping for High School: Algebra
Related Topics: Common Core for Mathematics Common Core Lesson Plans and Worksheets Algebra I
Algebra Overview
Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials Understand the relationship between zeros and factors of polynomials Use polynomial identities to solve problems Rewrite rational functions
Creating Equations Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically
Seeing Structure in Expressions
Arithmetic with Polynomials and Rational Expressions
Creating Equations
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.
Common Core State Standards Initiative
High School: Algebra
Standards in this domain:, ccss.math.content.hsa.introduction introduction, expressions..
An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.
Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05 p can be interpreted as the addition of a 5% tax to a price p . Rewriting p + 0.05 p as 1.05 p shows that adding a tax is the same as multiplying the price by a constant factor.
Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05 p is the sum of the simpler expressions p and 0.05 p . Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.
A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.
Equations and inequalities.
An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.
The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.
An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.
Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2 x + 1 = 0 is a rational number, not an integer; the solutions of x 2 - 2 = 0 are real numbers, not rational numbers; and the solutions of x 2 + 2 = 0 are complex numbers, not real numbers.
The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = (( b 1 + b 2 )/2) h , can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.
Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.
Algebra Overview
- Seeing Structure in Expressions
- Interpret the structure of expressions
- Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Functions
- Perform arithmetic operations on polynomials
- Understand the relationship between zeros and factors of polynomials
- Use polynomial identities to solve problems
- Rewrite rational functions
Creating Equations
- Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning
- Solve equations and inequalities in one variable
- Solve systems of equations
- Represent and solve equations and inequalities graphically
Mathematical Practices
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
Arithmetic with Polynomials & Rational Expressions
Creating equations ✭, reasoning with equations & inequalities.
- Standards for Mathematical Practice
- How to read the grade level standards
- Introduction
- Counting & Cardinality
- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Measurement & Data
- Number & Operations—Fractions¹
- Number & Operations in Base Ten¹
- Number & Operations—Fractions
- Ratios & Proportional Relationships
- The Number System
- Expressions & Equations
- Statistics & Probability
- The Real Number System
- Quantities*
- The Complex Number System
- Vector & Matrix Quantities
- Arithmetic with Polynomials & Rational Expressions
- Creating Equations*
- Reasoning with Equations & Inequalities
- Interpreting Functions
- Building Functions
- Linear, Quadratic, & Exponential Models*
- Trigonometric Functions
- High School: Modeling
- Similarity, Right Triangles, & Trigonometry
- Expressing Geometric Properties with Equations
- Geometric Measurement & Dimension
- Modeling with Geometry
- Interpreting Categorical & Quantitative Data
- Making Inferences & Justifying Conclusions
- Conditional Probability & the Rules of Probability
- Using Probability to Make Decisions
- Courses & Transitions
- Mathematics Glossary
- Mathematics Appendix A
IMAGES
VIDEO
COMMENTS
Students learn the basic method and reasoning behind graphing linear inequalities in two variables. For the worksheet used in this lesson, click on:http://ww...
Lesson 16: Graphing Quadratic Equations From the Vertex Form, y=a (x-h) 2 +k ( Vide0) Lesson 17: Graphing Quadratic Functions From the Standard Form, f (x)=ax 2 +bx+c ( Video Lesson) Function Transformations and Modeling. Topic C Overview. Lesson 18: Graphing Cubic, Square Root, and Cube Root Functions.
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!
Hello and welcome to another common core algebra one lesson. My name is Kirk Weiler, and today we're going to be doing unit four lesson number 11, graphs of linear inequalities. As a reminder, you can find the worksheet and a homework set that go along with this lesson by clicking on the video's description.
We develop general methods for solving linear equations using properties of equality and inverse operations. Thorough review is given to review of equation solving from Common Core 8th Grade Math. Solutions to equations and inequalities are defined in terms of making statements true. This theme is emphasized throughout the unit.
This unit begins by ensuring that students understand that solutions to equations are points that make the equation true, while solutions to systems make all equations (or inequalities) true. Graphical and substitution methods for solving systems are reviewed before the development of the Elimination Method.
Use a system of linear inequalities, of course. If it takes someone in a cooking competition 30 seconds to chop an onion and 18 seconds to chop a carrot, and their goal is to chop at least 20 of these vegetables in a time limit of 540 seconds, what can they do? Use a system of linear inequalities, of course.
We are so glad that you are taking responsibility for your education in Algebra I. SEMESTER 1. UNIT 1: Analyze Graphs and Expressions. UNIT 2: Create and Solve Equations/Ineq. UNIT 3: Systems of Equations. UNIT 4: One Variable Statistics. UNIT 5: Analyze Two Variable Data. UNIT 6: Sequences. SEMESTER 2.
A. Interpret parts of an expression, such as terms, factors, and coefficients. B. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P (1+r)n as the product of P and a factor not depending on P. Show Lessons. Show Worksheets/Games. HSA-SSE.A.2.
Find step-by-step solutions and answers to Algebra 1 Common Core - 9780133185485, as well as thousands of textbooks so you can move forward with confidence. ... Using Graphs to Relate Two Quantities. Section 4-2: Patterns and Linear Functions. Section 4-3: ... Systems of Linear Inequalities. Page 408: Chapter Review. Page 411: Chapter Test ...
CCSS.Math.Content.HSA.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 ...
CCSS.Math.Content.HSA.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.
Represent the solution set graphically. Create two-variable inequalities to represent a situation. Understand that a half-plane bounded by the line ax + by = c is a visual representation of the solution set to a linear inequality, such as ax + by < c. Interpret the inequality to determine which portion of the coordinate plane is shaded.
Section 2.11 : Linear Inequalities. Back to Problem List. 6. Solve the following inequality and give the solution in both inequality and interval notation. 2 < 1 6 − 1 2x ≤ 4 2 < 1 6 − 1 2 x ≤ 4. Show All Steps Hide All Steps.
a. For a quadratic function, use an algebraic process to find zeros, maxima, minima, and symmetry of the graph, and interpret these in terms of context. Algebraic processes include but not limited to factoring, completing the square, use of the quadratic formula, and the use of the axis of symmetry. F.IF.C.9.
Unit 4 - Linear Functions and Arithmetic Sequences. This unit is all about understanding linear functions and using them to model real world scenarios. Fluency in interpreting the parameters of linear functions is emphasized as well as setting up linear functions to model a variety of situations. Linear inequalities are also taught.
Recall that if a line is written in the form y mx b , then it is relatively easy to graph, especially if m and b are reasonably easy to work with. A quick review from the previous lesson. Exercise #2: On the grid below, graph the equation 3 3 2 yx . First, identify its slope and y-intercept to help you with the graph. Slope: _____
3. hw 6.3 systems of inequalities. Hw solving systems of linear inequalities. Common core algebra i.unit 5.lesson 8.modeling with systems of inequalities by emathinstruction. Day 13 graphing inequalities, computer lab desmos graphing, warm up, homework graphing absolute value/piecewise functions, classwork computer lab.
Common Core Standard: A-REI.C.6, A-REI.D.10, and A-REI.D.12. Student Outcome: Students will solve systems of linear equations graphically, understanding there are three different types of solutions. Students will also use technology (graphing calculator) to solve a system by graphing.
Standards Alignment - Powered by EdGate. Table of Contents for Common Core Algebra I. Unit 1 - The Building Blocks of Algebra. Unit 2 - Linear Expressions, Equations, and Inequalities. Unit 3 - Functions. Unit 4 - Linear Functions and Arithmetic Sequences. Unit 5 - Systems of Linear Equations and Inequalities.
File Type: pdf. Download File. Units 0, 1, & 2 packets are free! To purchase the entire course of lesson packets, click here. Practice Solutions. alg_2.3_practice_solutions_.pdf. File Size: 844 kb.
Unit 8 - Quadratic Functions and Their Algebra. These lessons introduce quadratic polynomials from a basic perspective. We then build on the notion of shifting basic parabolas into their vertex form. Completing the square is used as a fundamental tool in finding the turning point of a parabola. Finally, the zero product law is introduced as a ...
COMMON CORE ALGEBRA I. Linear inequalities tend to have an infinite amount of values for the replacement variable (typically x) that solve the inequality. Sometimes, we put two (or more) inequalities together and ask what x values make both true (AND) and which make either one or the other true (OR). You will deal with AND and OR along with truth.