# Algebra 1 Systems of Inequalities

Section 6: Inequalities: Lecture 6 | 21:13 min

In this lesson we are going to wrap things up with inequalities by looking at systems of inequalities. We will look at how you can satisfy a system of inequalities rather just individual inequalities. That will bring us to looking at overlapping solution regions. To graph a system of inequalities, graph each inequality separately. The solution region of the system is where the individual graphs overlap. You may use a test point from the overlapping region in the original system to see if the solution region is correct. Look for word problems that involve a range of possibilities to determine if an inequality would be more appropriate when setting it up.

Eric Smith

Systems of Inequalities

Slide Duration:Table of Contents

30m 41s

- Intro0:00
- Objectives0:07
- Basic Types of Numbers0:36
- Natural Numbers1:02
- Whole Numbers1:29
- Integers2:04
- Rational Numbers2:38
- Irrational Numbers5:06
- Imaginary Numbers6:48
- Basic Types of Numbers Cont.8:09
- The Big Picture8:10
- Real vs. Imaginary Numbers8:30
- Rational vs. Irrational Numbers8:48
- Basic Types of Numbers Cont.10:55
- Number Line11:06
- Absolute Value11:44
- Inequalities12:39
- Example 113:16
- Example 217:30
- Example 321:56
- Example 424:27
- Example 527:48

19m 26s

- Intro0:00
- Objectives0:06
- Operations on Numbers0:25
- Addition0:53
- Subtraction1:33
- Multiplication & Division2:19
- Exponents3:24
- Bases4:04
- Square Roots4:59
- Principle Square Roots5:09
- Perfect Squares6:32
- Simplifying and Combining Roots6:52
- Example 18:16
- Example 212:30
- Example 314:02
- Example 416:27

12m 6s

- Intro0:00
- Objectives0:06
- The Order of Operations0:25
- Work Inside Parentheses0:42
- Simplify Exponents0:52
- Multiplication & Division from Left to Right0:57
- Addition & Subtraction from Left to Right1:11
- Remember PEMDAS1:21
- The Order of Operations Cont.2:27
- Example2:43
- Example 13:55
- Example 25:36
- Example 37:35
- Example 48:56

18m 52s

- Intro0:00
- Objectives0:07
- The Properties of Real Numbers0:23
- Commutative Property of Addition and Multiplication0:44
- Associative Property of Addition and Multiplication1:50
- Distributive Property of Multiplication Over Addition3:20
- Division Property of Zero4:46
- Division Property of One5:23
- Multiplication Property of Zero5:56
- Multiplication Property of One6:17
- Addition Property of Zero6:29
- Why Are These Properties Important?6:53
- Example 19:16
- Example 213:04
- Example 314:30
- Example 416:57

12m 22s

- Intro0:00
- Objectives0:09
- The Vocabulary of Linear Equations0:44
- Variables0:52
- Terms1:09
- Coefficients1:40
- Like Terms2:18
- Examples of Like Terms2:37
- Expressions4:01
- Equations4:26
- Linear Equations5:04
- Solutions5:55
- Example 16:16
- Example 27:16
- Example 38:45
- Example 410:20

28m 52s

- Intro0:00
- Objectives0:08
- Solving Linear Equations in One Variable0:34
- Conditional Cases0:51
- Identity Cases1:09
- Contradiction Cases1:30
- Solving Linear Equations in One Variable Cont.2:00
- Addition Property of Equality2:10
- Multiplication Property of Equality2:43
- Steps to Solve Linear Equations3:14
- Example 14:22
- Example 28:21
- Example 312:32
- Example 414:19
- Example 517:25
- Example 622:17

12m 2s

- Intro0:00
- Objectives0:06
- Solving Formulas0:18
- Formulas0:26
- Use the Same Properties as Solving Linear Equations1:36
- Addition Property of Equality1:55
- Multiplication Property of Equality1:58
- Steps to Solve Formulas2:43
- Example 13:56
- Example 26:09
- Example 38:39

28m 41s

- Intro0:00
- Objectives0:10
- Applications of Linear Equations0:43
- The Six-Step Method to Solving Word Problems0:55
- Common Terms3:12
- Example 15:03
- Example 29:40
- Example 313:48
- Example 417:58
- Example 523:28

24m 26s

- Intro0:00
- Objectives0:21
- Motion and Mixtures0:46
- Motion Problems: Distance, Rate, and Time1:06
- Mixture Problems: Amount, Percent, and Total1:27
- The Table Method1:58
- The Beaker Method3:38
- Example 15:05
- Example 29:44
- Example 314:20
- Example 419:13

22m 55s

- Intro0:00
- Objectives0:11
- The Rectangular Coordinate System0:39
- The Cartesian Coordinate System0:40
- X-Axis0:54
- Y-Axis1:04
- Origin1:11
- Quadrants1:26
- Ordered Pairs2:10
- Example 12:55
- The Rectangular Coordinate System Cont.6:09
- X-Intercept6:45
- Y-Intercept6:55
- Relation of X-Values and Y-Values7:30
- Example 211:03
- Example 312:13
- Example 414:10
- Example 518:38

27m 58s

- Intro0:00
- Objectives0:11
- Slope and Graphing0:48
- Standard Form1:14
- Example 12:24
- Slope and Graphing Cont.4:58
- Slope, m5:07
- Slope is Rise over Run6:11
- Don't Mix Up the Coordinates8:20
- Example 29:39
- Slope and Graphing Cont.14:26
- Slope-Intercept Form14:34
- Example 316:55
- Example 418:00
- Slope and Graphing Cont.19:00
- Rewriting an Equation in Slope-Intercept Form19:39
- Rewriting an Equation in Standard Form20:09
- Slopes of Vertical & Horizontal Lines20:56
- Example 522:49
- Example 624:09
- Example 725:59
- Example 826:57

20m 36s

- Intro0:00
- Objectives0:13
- Linear Equations in Two Variables0:36
- Point-Slope Form1:07
- Substitute in the Point and the Slope2:21
- Parallel Lines: Two Lines with the Same Slope4:05
- Perpendicular Lines: Slopes are Negative Reciprocals of Each Other4:39
- Perpendicular Lines: Product of Slopes is -15:24
- Example 16:02
- Example 27:50
- Example 310:49
- Example 413:26
- Example 515:30
- Example 617:43

21m 24s

- Intro0:00
- Objectives0:07
- Introduction to Functions0:58
- Relations1:03
- Functions1:37
- Independent Variables2:00
- Dependent Variables2:11
- Function Notation2:21
- Function3:43
- Input and Output3:53
- Introduction to Functions Cont.4:45
- Domain4:46
- Range4:55
- Functions Represented by a Diagram6:41
- Natural Domain9:11
- Evaluating Functions12:02
- Example 113:13
- Example 215:03
- Example 316:18
- Example 419:54

16m 12s

- Intro0:00
- Objectives0:09
- Graphing Functions0:54
- Using Slope-Intercept Form1:56
- Vertical Line Test2:58
- Determining the Domain4:20
- Determining the Range5:43
- Example 16:06
- Example 27:18
- Example 38:31
- Example 411:04

25m 54s

- Intro0:00
- Objectives0:13
- Systems of Linear Equations0:46
- System of Equations0:51
- System of Linear Equations1:15
- Solutions1:35
- Points as Solutions1:53
- Finding Solutions Graphically5:13
- Example 16:37
- Example 212:07
- Systems of Linear Equations Cont.17:01
- One Solution, No Solution, or Infinite Solutions17:10
- Example 318:31
- Example 422:37

20m 1s

- Intro0:00
- Objectives0:09
- Solving a System Using Substitution0:32
- Substitution Method1:24
- Substitution Example2:35
- One Solution, No Solution, or Infinite Solutions7:50
- Example 19:45
- Example 212:48
- Example 315:01
- Example 417:30

19m 40s

- Intro0:00
- Objectives0:09
- Solving a System Using Elimination0:27
- Elimination Method0:42
- Elimination Example2:01
- One Solution, No Solution, or Infinite Solutions7:05
- Example 18:53
- Example 211:46
- Example 315:37
- Example 417:45

24m 34s

- Intro0:00
- Objectives0:12
- Applications of Systems of Equations0:30
- Word Problems1:31
- Example 12:17
- Example 27:55
- Example 313:07
- Example 417:15

17m 13s

- Intro0:00
- Objectives0:08
- Solving Linear Inequalities in One Variable0:37
- Inequality Expressions0:46
- Linear Inequality Solution Notations3:40
- Inequalities3:51
- Interval Notation4:04
- Number Lines4:43
- Set Builder Notation5:24
- Use Same Techniques as Solving Equations6:59
- 'Flip' the Sign when Multiplying or Dividing by a Negative Number7:12
- 'Flip' Example7:50
- Example 18:54
- Example 211:40
- Example 314:01

16m 13s

- Intro0:00
- Objectives0:07
- Compound Inequalities0:37
- 'And' vs. 'Or'0:44
- 'And'3:24
- 'Or'3:35
- 'And' Symbol, or Intersection3:51
- 'Or' Symbol, or Union4:13
- Inequalities4:41
- Example 16:22
- Example 29:30
- Example 311:27
- Example 413:49

14m 12s

- Intro0:00
- Objectives0:08
- Solve Equations with Absolute Values0:18
- Solve Equations with Absolute Values Cont.1:11
- Steps to Solving Equations with Absolute Values2:21
- Example 13:23
- Example 26:34
- Example 310:12

17m 7s

- Intro0:00
- Objectives0:07
- Inequalities with Absolute Values0:23
- Recall…2:08
- Example 13:39
- Example 26:06
- Example 38:14
- Example 410:29
- Example 513:29

15m 33s

- Intro0:00
- Objectives0:07
- Graphing Inequalities in Two Variables0:32
- Split Graph into Two Regions1:53
- Graphing Inequalities5:44
- Test Points6:20
- Example 17:11
- Example 210:17
- Example 313:06

21m 13s

- Intro0:00
- Objectives0:08
- Systems of Inequalities0:24
- Test Points1:10
- Steps to Solve Systems of Inequalities1:25
- Example 12:23
- Example 27:28
- Example 312:51

44m 51s

- Intro0:00
- Objectives0:09
- Integer Exponents0:42
- Exponents 'Package' Multiplication1:25
- Example 12:00
- Example 23:13
- Integer Exponents Cont.4:50
- Product Rule for Exponents4:51
- Example 37:16
- Example 410:15
- Integer Exponents Cont.13:13
- Power Rule for Exponents13:14
- Power Rule with Multiplication and Division15:33
- Example 516:18
- Integer Exponents Cont.20:04
- Example 620:41
- Integer Exponents Cont.25:52
- Zero Exponent Rule25:53
- Quotient Rule28:24
- Negative Exponents30:14
- Negative Exponent Rule32:27
- Example 734:05
- Example 836:15
- Example 939:33
- Example 1043:16

18m 33s

- Intro0:00
- Objectives0:07
- Adding and Subtracting Polynomials0:25
- Terms0:33
- Coefficients0:51
- Leading Coefficients1:13
- Like Terms1:29
- Polynomials2:21
- Monomials, Binomials, Trinomials, and Polynomials5:41
- Degrees7:00
- Evaluating Polynomials8:12
- Adding and Subtracting Polynomials Cont.9:25
- Example 111:48
- Example 213:00
- Example 314:41
- Example 416:15

25m 7s

- Intro0:00
- Objectives0:06
- Multiplying Polynomials0:41
- Distributive Property1:00
- Example 12:49
- Multiplying Polynomials Cont.8:22
- Organize Terms with a Table8:23
- Example 213:40
- Multiplying Polynomials Cont.16:33
- Multiplying Binomials with FOIL16:48
- Example 318:49
- Example 420:04
- Example 521:42

44m 56s

- Intro0:00
- Objectives0:07
- Dividing Polynomials0:29
- Dividing Polynomials by Monomials2:10
- Dividing Polynomials by Polynomials2:59
- Dividing Numbers4:09
- Dividing Polynomials Example8:39
- Example 112:35
- Example 214:40
- Example 316:45
- Example 421:13
- Example 524:33
- Example 629:02
- Dividing Polynomials with Synthetic Division Method33:36
- Example 738:43
- Example 842:24

28m 27s

- Intro0:00
- Objectives0:09
- Greatest Common Factor0:31
- Factoring0:40
- Greatest Common Factor (GCF)1:48
- GCF for Polynomials3:28
- Factoring Polynomials6:45
- Prime8:21
- Example 19:14
- Factor by Grouping14:30
- Steps to Factor by Grouping17:03
- Example 217:43
- Example 319:20
- Example 420:41
- Example 522:29
- Example 626:11

21m 44s

- Intro0:00
- Objectives0:06
- Factoring Trinomials0:25
- Recall FOIL0:26
- Factor a Trinomial by Reversing FOIL1:52
- Tips when Using Reverse FOIL5:31
- Example 17:04
- Example 29:09
- Example 311:15
- Example 413:41
- Factoring Trinomials Cont.15:50
- Example 518:42

30m 9s

- Intro0:00
- Objectives0:08
- Factoring Trinomials Using the AC Method0:27
- Factoring when Leading Term has Coefficient Other Than 11:07
- Reversing FOIL1:18
- Example 11:46
- Example 24:28
- Factoring Trinomials Using the AC Method Cont.7:45
- The AC Method8:03
- Steps to Using the AC Method8:19
- Tips on Using the AC Method9:29
- Example 310:45
- Example 416:50
- Example 521:08
- Example 624:58

30m 14s

- Intro0:00
- Objectives0:07
- Special Factoring Techniques0:26
- Difference of Squares1:46
- Perfect Square Trinomials2:38
- No Sum of Squares3:32
- Special Factoring Techniques Cont.4:03
- Difference of Squares Example4:04
- Perfect Square Trinomials Example5:29
- Example 17:31
- Example 29:59
- Example 311:47
- Example 415:09
- Special Factoring Techniques Cont.19:07
- Sum of Cubes and Difference of Cubes19:08
- Example 523:13
- Example 626:12

23m 38s

- Intro0:00
- Objectives0:08
- Solving Quadratic Equations by Factoring0:19
- Quadratic Equations0:20
- Zero Factor Property1:39
- Zero Factor Property Example2:34
- Example 14:00
- Solving Quadratic Equations by Factoring Cont.5:54
- Example 27:28
- Example 311:09
- Example 414:22
- Solving Quadratic Equations by Factoring Cont.18:17
- Higher Degree Polynomial Equations18:18
- Example 520:22

29m 27s

- Intro0:00
- Objectives0:12
- Solving Quadratic Equations0:29
- Linear Factors0:38
- Not All Quadratics Factor Easily1:22
- Principle of Square Roots3:36
- Completing the Square4:50
- Steps for Using Completing the Square5:15
- Completing the Square Works on All Quadratic Equations6:41
- The Quadratic Formula7:28
- Discriminants8:25
- Solving Quadratic Equations - Summary10:11
- Example 111:54
- Example 213:03
- Example 316:30
- Example 421:29
- Example 525:07

16m 47s

- Intro0:00
- Objectives0:08
- Equations in Quadratic Form0:24
- Using a Substitution0:53
- U-Substitution1:26
- Example 12:07
- Example 25:36
- Example 38:31
- Example 411:14

29m 4s

- Intro0:00
- Objectives0:09
- Quadratic Formulas and Applications0:35
- Squared Variable0:40
- Principle of Square Roots0:51
- Example 11:09
- Example 22:04
- Quadratic Formulas and Applications Cont.3:34
- Example 34:42
- Example 413:33
- Example 520:50

26m 53s

- Intro0:00
- Objectives0:06
- Graphs of Quadratics0:39
- Axis of Symmetry1:46
- Vertex2:12
- Transformations2:57
- Graphing in Quadratic Standard Form3:23
- Example 15:06
- Example 26:02
- Example 39:07
- Graphs of Quadratics Cont.11:26
- Completing the Square12:02
- Vertex Shortcut12:16
- Example 413:49
- Example 517:25
- Example 620:07
- Example 723:43

21m 42s

- Intro0:00
- Objectives0:07
- Polynomial Inequalities0:30
- Solving Polynomial Inequalities1:20
- Example 12:45
- Polynomial Inequalities Cont.5:12
- Larger Polynomials5:13
- Positive or Negative Intervals7:16
- Example 29:01
- Example 313:53

26m 41s

- Intro0:00
- Objectives0:09
- Multiply and Divide Rational Expressions0:44
- Rational Numbers0:55
- Dividing by Zero1:45
- Canceling Extra Factors2:43
- Negative Signs in Fractions4:52
- Multiplying Fractions6:26
- Dividing Fractions7:17
- Example 18:04
- Example 214:01
- Example 316:23
- Example 418:56
- Example 522:43

20m 24s

- Intro0:00
- Objectives0:07
- Adding and Subtracting Rational Expressions0:41
- Common Denominators0:52
- Common Denominator Examples1:14
- Steps to Adding and Subtracting Rational Expressions2:39
- Example 13:34
- Example 25:27
- Adding and Subtracting Rational Expressions Cont.6:57
- Least Common Denominators6:58
- Transitioning from Fractions to Rational Expressions9:08
- Identifying Least Common Denominators for Rational Expressions9:56
- Subtracting vs. Adding10:41
- Example 311:19
- Example 412:36
- Example 515:08
- Example 616:46

18m 23s

- Intro0:00
- Objectives0:09
- Complex Fractions0:37
- Dividing to Simplify Complex Fractions1:10
- Example 12:03
- Example 23:58
- Complex Fractions Cont.9:15
- Using the Least Common Denominator to Simplify Complex Fractions9:16
- Both Methods Lead to the Same Answer10:07
- Example 310:42
- Example 414:28

16m 24s

- Intro0:00
- Objectives0:07
- Solving Rational Equations0:23
- Isolate the Specified Variable1:23
- Example 11:58
- Example 25:00
- Example 38:23
- Example 413:25

18m 54s

- Intro0:00
- Objectives0:06
- Rational Inequalities0:18
- Testing Intervals for Rational Inequalities0:38
- Steps to Solving Rational Inequalities1:05
- Tips to Solving Rational Inequalities2:27
- Example 13:33
- Example 212:21

20m 20s

- Intro0:00
- Objectives0:07
- Applications of Rational Expressions0:27
- Work Problems1:05
- Example 12:58
- Example 26:45
- Example 313:17
- Example 416:37

27m 4s

- Intro0:00
- Objectives0:10
- Variation and Proportion0:34
- Variation0:35
- Inverse Variation1:01
- Direct Variation1:10
- Setting Up Proportions1:31
- Example 12:27
- Example 25:36
- Variation and Proportion Cont.8:29
- Inverse Variation8:30
- Example 39:20
- Variation and Proportion Cont.12:41
- Constant of Proportionality12:42
- Example 413:59
- Variation and Proportion Cont.16:17
- Varies Directly as the nth Power16:30
- Varies Inversely as the nth Power16:53
- Varies Jointly17:09
- Combining Variation Models17:36
- Example 519:09
- Example 622:10

14m 32s

- Intro0:00
- Objectives0:07
- Rational Exponents0:32
- Power on Top, Root on Bottom1:05
- Example 11:37
- Rational Exponents Cont.4:04
- Using Rules from Exponents for Radicals as Exponents4:05
- Combining Terms Under a Single Root4:50
- Example 25:21
- Example 37:39
- Example 411:23
- Example 513:14

15m 12s

- Intro0:00
- Objectives0:07
- Simplify Rational Exponents0:25
- Product Rule for Radicals0:26
- Product Rule to Simplify Square Roots1:11
- Quotient Rule for Radicals1:42
- Applications of Product and Quotient Rules2:17
- Higher Roots2:48
- Example 13:39
- Example 26:35
- Example 38:41
- Example 411:09

17m 22s

- Intro0:00
- Objectives0:07
- Adding and Subtracting Radicals0:33
- Like Terms1:29
- Bases and Exponents May be Different2:02
- Bases and Powers Must be Same when Adding and Subtracting2:42
- Add Radicals' Coefficients3:55
- Example 14:47
- Example 26:00
- Adding and Subtracting Radicals Cont.7:10
- Simplify the Bases to Look the Same7:25
- Example 38:23
- Example 411:45
- Example 515:10

19m 24s

- Intro0:00
- Objectives0:08
- Multiply and Divide Radicals0:25
- Rules for Working With Radicals0:26
- Using FOIL for Radicals1:11
- Don’t Distribute Powers2:54
- Dividing Radical Expressions4:25
- Rationalizing Denominators6:40
- Example 17:22
- Example 28:32
- Multiply and Divide Radicals Cont.9:23
- Rationalizing Denominators with Higher Roots9:25
- Example 310:51
- Example 411:53
- Multiply and Divide Radicals Cont.13:13
- Rationalizing Denominators with Conjugates13:14
- Example 515:52
- Example 617:25

15m 5s

- Intro0:00
- Objectives0:07
- Solving Radical Equations0:17
- Radical Equations0:18
- Isolate the Roots and Raise to Power0:34
- Example 11:13
- Example 23:09
- Solving Radical Equations Cont.7:04
- Solving Radical Equations with More than One Radical7:05
- Example 37:54
- Example 413:07

29m 16s

- Intro0:00
- Objectives0:06
- Complex Numbers1:05
- Imaginary Numbers1:08
- Complex Numbers2:27
- Real Parts2:48
- Imaginary Parts2:51
- Commutative, Associative, and Distributive Properties3:35
- Adding and Subtracting Complex Numbers4:04
- Multiplying Complex Numbers6:16
- Dividing Complex Numbers8:59
- Complex Conjugate9:07
- Simplifying Powers of i14:34
- Shortcut for Simplifying Powers of i18:33
- Example 121:14
- Example 222:15
- Example 323:38
- Example 426:33

For more information, please see full course syllabus of Algebra 1

2 answers

Last reply by: Sage Stark

Mon Jul 24, 2017 2:41 PM

Post by Sage Stark on July 24, 2017

Is there any algebraic way of solving systems of inequalities, or is graphing the only way?

1 answer

Last reply by: Professor Eric Smith

Sat Aug 10, 2013 1:05 PM

Post by Emily Engle on August 10, 2013

How would you find the numeric values of the solution of the inequalities ?