Back To Course

High School Algebra II: Help and Review26 chapters | 296 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Glenda Boozer*

When we need to solve a system of two or more inequalities, we graph the solution set of each one, but we put them on the same graph and see what that looks like. The overlapping shaded areas tells us where to find the solution set.

When we have a system of two or more inequalities to solve, we will graph the solution set of each one, but we will put them on the same graph and see what that looks like. The overlapping shaded areas will tell us where to find the solution set. For example, let's try this system:

*y* > *x*^2

*y* <= *x* + 1

We will start by choosing one of the inequalities and graphing the equation associated with it. An **equation** is a statement that two quantities are equal. We can create that equation by replacing the >, <, >=, or <= signs by an = sign. For example, if we have *y* > *x*^2, we'll graph the equation *y* = *x*^2:

We used a dotted line here instead of a solid line because points along the line are not included in the **solution set**, or the set of all solutions to the equation. How do we know? Let's take the point (2,4). In our inequality we replace *x* with 2 and *y* with 4, and we have 4 > 2^2; but 4 is not greater than 2^2, it is equal. Therefore, this point is not part of the solution set.

We have only made the boundary of the solution set so far; we need the graph to show all of the points where *y* is greater than x^2. Let's choose a point on the graph and see if it is part of the solution set. We'll try the point (0,1) since it is definitely above the dotted line. In fact, if it is part of the solution set, everything above the dotted line will be part of the solution set. Once more, we substitute 0 (the first number) for *x*, and the 1 (the second number) for *y* in the inequality, and we have 1 > 0^2. Hey, that's true! We can shade the area above the line:

Next, we graph the line *y* = x + 1 on the same grid. We will use a solid line here, because the inequality is *y* <= *x* + 1 and not just *y* < *x* + 1. You can see it in the graph as a straight, solid red line. Which side of the line should we shade? Let's test the point (0,0) and see if its *x* and *y* values will make the inequality true. We get 0 <= 0 + 1; this is true, so we shade the side that includes the point (0,0). This graph uses pale blue shading for the solution set of *y* <= *x* +1 to make it easier to see the solution sets of both inequalities.

If we had three ineualities, or four, or any number, we would graph them on the same grid, using the same method.

Now that everything is graphed and shaded, we can use our graph to find our final solution. The area where the pink shading and the pale blue shading overlap, between the solid line and the dotted curve, is the solution set for both *y* > *x*^2 and *y* <= *x* + 1. Notice that, while the point (0,0) was part of the solution set for *y* <= *x* + 1, it is not part of the solution set for *y* > *x*^2; we can see that because it is on a dotted line and not a solid one, and it is not completely inside the area with the overlapping shading.

On the other hand, the point (0,1) is just fine; it is on a solid line that touches the double-shaded area. So is the point (1,2). If we picture the point (0,0.5), we can see that it is completely inside the shaded area. We can plot any point and tell whether it is part of the solution set or not.

What if the shading was in two completely separate places with no overlap? Then the solution set would be the **empty set**, a set with no elements.

Whenever we want to find the solution set for a system of inequalities by graphing, we can use the following strategy:

- Graph the equation associated with one of the inequalities. Use a solid line if the inequality is >= or <=, and use a dotted line if it is > or <.
- Determine which side of the line represents the solution set by testing a point that is not directly on the line. If we can replace
*x*with the first coordinate and*y*with the second coordinate in the inequality and make a true statement, then shade the side of the line where the point is located. If not, shade the other side. - Repeat steps 1 and 2 until all of the inequalities are graphed and shaded on the same grid. The region of the graph where all of the shading overlaps represents the solution set. If the shaded areas do not all overlap, the solution set is
**empty**.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
8 in chapter 9 of the course:

Back To Course

High School Algebra II: Help and Review26 chapters | 296 lessons

- What is an Inequality? 7:09
- How to Graph 1- and 2-Variable Inequalities 7:59
- Set Notation, Compound Inequalities, and Systems of Inequalities 8:16
- Graphing Inequalities: Practice Problems 12:06
- How to Solve and Graph an Absolute Value Inequality 8:02
- Solving and Graphing Absolute Value Inequalities: Practice Problems 9:06
- Translating Math Sentences to Inequalities 5:36
- System of Inequalities: Graphing & Concept 4:58
- Go to Algebra II - Inequalities Review: Help and Review

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- What Are the 5 Ws in Writing? - Uses & Examples
- Phenol: Preparation & Reactions
- What is a Color Wheel? - Definition & Types
- What Are Abbreviations? - Meaning, Types & Examples
- Zentangle Lesson Plan for High School
- West Side Story Discussion Questions
- Fireboat: The Heroic Adventures of the John J. Harvey Activities
- Quiz & Worksheet - Solvay Process
- Quiz & Worksheet - Acetone Reactions
- Quiz & Worksheet - Themes in A Raisin in the Sun
- Quiz & Worksheet - Act & Rule Utilitarianism Comparison
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Math Worksheets | Printable Math Worksheets for Teachers
- Calculus Worksheets

- UExcel Foundations of Gerontology: Study Guide & Test Prep
- 12th Grade English: Homework Help Resource
- Business Law: Skills Development & Training
- High School Psychology: Homeschool Curriculum
- History of the Vietnam War for Teachers: Professional Development
- MTTC History: Hinduism
- Emotions & Emotional Expression
- Quiz & Worksheet - Characteristics of Curriculum Design
- Quiz & Worksheet - Features of Early Childhood Curricula
- Quiz & Worksheet - How to Teach Reading to ELL Students
- Quiz & Worksheet - Starbucks & Business Ethics
- Quiz & Worksheet - Chomsky's Language Acquisition Device

- Psychometrics: Definition & Test Examples
- Jigsaw Writing Activities
- First Grade Word Walls: List & Activities
- Constitutional Convention Lesson Plan
- Listening Activities for Kids
- Multiplication Rhymes & Tricks for Kids
- Homeschooling in New Jersey
- Wisconsin State Social Studies Standards
- Civil War Activities for Kids
- Curriculum-Based Assessment Examples
- Suicide Prevention Grants
- Communism Lesson Plan

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject