Slide 1Inequalities and ApplicationsSlide 3Slide 4Which numbers would be solutions of ?Representing InequalitiesSlide 7Slide 8Slide 9Slide 10Slide 11Slide 12Write in interval notation and graph on a number line.Write using set-builder notation and using interval notation.Write using set-builder notation and graph on a number line.Slide 16Slide 17Slide 18Begin with 20 > 12Slide 20Example 2 from lesson notesSlide 22Example 3 from lesson notesExample 4 Lesson NotesSlide 25Example 5 from lesson notesExample 6 from lesson notesExample 7 from lesson notesExample 8Example 9 from lesson notesSlide 31Application Example 1Application Example 2Application Example 3Application Example 4Application Example 5Application Example 6Slide 4- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyCopyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyInequalities and ApplicationsSolving InequalitiesInterval NotationThe Addition Principle for InequalitiesThe Multiplication Principle for InequalitiesUsing the Principles TogetherProblem Solving4.1Slide 4- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleySolving InequalitiesAn inequality is any sentence containing , , , , or .< > � � �Any value for a variable that makes an inequality true is called a solution. The set of all solutions is called the solution set. When all solutions of an inequality are found, we say that we have solved the inequality.Examples3 2 7, 7, and 4 6 3.x c x+ > � - �Slide 4- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyExampleSolutionDetermine whether 5 is a solution to 3 2 7.x + >We substitute to get 3(5) + 2 > 7, or 17 >7, a true statement. Thus, 5 is a solution.Slide 4- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyWhich numbers would be solutions of ?4 5 2( 1)x x+ > -0, 5, 3, 12- -T F T TSlide 4- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyRepresenting InequalitiesInequality Symbol x < 2Number Line Graph Interval Notation-5[3,7)Slide 4- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyThe graph of an inequality is a visual representation of the inequality’s solution set. An inequality in one variable can be graphed on a number line.ExampleGraph x < 2 on a number line.Solution-5 -4 -3 -2 -1 0 1 2 3 4 5 6 Note that in set-builder notation the solution is{ }| 2 .x x <Slide 4- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyInterval Notation Another way to write solutions of an inequality in one variable is to use interval notation. Interval notation uses parentheses, ( ), and brackets, [ ].If a and b are real numbers such that a < b, we define the open interval (a, b) as the set of all numbers x for which a < x < b. Thus, { }( , ) | .a b x a x b= < <a b(a, b)Slide 4- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyInterval Notation The closed interval [a, b] is defined as the set of all numbers x for which Thus, { }[ , ] | .a b x a x b= � �.a x b� �a b[a, b]Slide 4- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyInterval Notation There are two types of half-open intervals, defined as follows: { }1. ( , ] | .a b x a x b= < �{ }2. [ , ) | .a b x a x b= � <a b[a, b)(a, b]a bSlide 4- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyInterval Notation We use the symbols to represent positive and negative infinity, respectively. Thus the notation (a, ) represents the set of all real numbers greater than a, and ( , a) represents the set of all numbers less than a.�- � and � - �( , )a- �( , )a �aaThe notations (– , a] and [a, ) are used when we want to include the endpoint a.��Slide 4- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley{ | 5} ( ,5){ | 2} [2, ){ | 0 3} x xx xx x< - �� �< < (0,3){ | 4 10} [4,10]{ | 3 1} ( 3,1]x xx x� �- < � - 520 3410-3 1Slide 4- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyWrite in interval notation and graph on a number line.{ | 2}y y �( , 2]- �2Slide 4- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyWrite using set-builder notation and using interval notation.{ | 5}x x >--5( 5, )- �Slide 4- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyWrite using set-builder notation and graph on a number line.[0,5){ | 0 5}x x� <0 5Slide 4- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyThe Addition Principle for InequalitiesTwo inequalities are equivalent if they have the same solution set. For example, the inequalities x > 3 and 3 < x are equivalent. Just as the addition principle for equations produces equivalent equations, the addition principle for inequalities produces equivalent inequalities.Slide 4- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyThe Addition Principle for InequalitiesFor any real numbers a, b, and c:a < b is equivalent to a + c < b + c;a > b is equivalent to a + c > b + c;Similar statements hold for and .� �Since subtraction of c is the same as addition by -c, there is no need for a separate subtraction principle.Slide 4- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyThe Multiplication Principle for InequalitiesFor any real numbers a, b, and for any positive number c:a < b is equivalent to ac < bc;a > b is equivalent to ac > bc.For any real numbers a, b, and for any negative number c:a < b is equivalent to ac > bc;a > b is equivalent to ac < bc.Similar statements hold for (for division too.) and .� �Slide 4- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyBegin with 20 > 12Add 4 to both sides: 20 + 4 > 12 + 4 trueSubtract 3 from both sides: 20 – 3 > 12 – 3 trueMultiply by 4: 4(20) > 12(20) trueDivide by 2: trueMultiply by -2: -2(20) >-2(12) falseDivide by -4: false 212220412420Slide 4- 20 Copyright © 2006 Pearson Education, Inc.
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