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Intermediate Algebra Exam 3 Material Inequalities and Absolute Value Inequalities • An equation is a comparison that says two algebraic expressions are equal • An inequality is a comparison between two or three algebraic expressions using symbols for: greater than: greater than or equal to: less than: less than or equal to: • Examples: x 15 3x 3 Two part inequality 3 x 4 1 1 Three part inequality . 2 Inequalities • There are lots of different types of inequalities, and each is solved in a special way • Inequalities are called equivalent if they have exactly the same solutions • Equivalent inequalities are obtained by using “properties of inequalities” Properties of Inequalities • Adding or subtracting the same number to all parts of an inequality gives an equivalent inequality with the same sense (direction) of the inequality symbol Add 3 x 3 2 is equivalent to : x 5 • Multiplying or dividing all parts of an inequality by the same POSITIVE number gives an equivalent inequality with the same sense (direction) of the inequality symbol Divide by 3 3x 6 is equivalent to : x 2 • Multiplying or dividing all parts of an inequality by the same NEGATIVE number and changing the sense (direction) of the inequality symbol gives an equivalent inequality Divide by - 2 2 x 8 is equivalent to : x 4 Solutions to Inequalities • Whereas solutions to equations are usually sets of individual numbers, solutions to inequalities are typically intervals of numbers • Example: Solution to x = 3 is {3} Solution to x < 3 is every real number that is less than three • Solutions to inequalities may be expressed in: – Standard Notation – Graphical Notation – Interval Notation Two Part Linear Inequalities • A two part linear inequality is one that looks the same as a linear equation except that an equal sign is replaced by inequality symbol (greater than, greater than or equal to, less than, or less than or equal to) • Example: x 15 3x 3 Expressing Solutions to Two Part Inequalities • “Standard notation” - variable appears alone on left side of inequality symbol, and a number appears alone on right side: x2 • “Graphical notation” - solutions are shaded on a number line using arrows to indicate all numbers to left or right of where shading ends, and using a parenthesis to indicate that a number is not included, and a square bracket to indicate that a number is included ]2 • “Interval notation” - solutions are indicated by listing in order the smallest and largest numbers that are in the solution interval, separated by comma, enclosed within parenthesis and/or square bracket. If there is no limit in the negative direction, “negative infinity symbol” is used, and if there is no limit in the positive direction, a “positive infinity symbol” is used. When infinity symbols are used, they are always used with a parenthesis. (, 2] Solving Two Part Linear Inequalities • Solve exactly like linear equations EXCEPT: – Always isolate variable on left side of inequality – Correctly apply principles of inequalities (In particular, always remember to reverse sense of inequality when multiplying or dividing by a negative) Example of Solving Two Part Linear Inequalities x 15 3x 3 x 15 3x 9 2x 6 When dividing by a negative, reversesense of inequality ! x 3 Standard Notation Solution 3 ] Graphical Notation Solution (, 3] Interval Notation Solution Three Part Linear Inequalities • Consist of three algebraic expressions compared with two inequality symbols • Both inequality symbols MUST have the same sense (point the same direction) AND must make a true statement when the middle expression is ignored • Good Example: 3 x 4 1 1 2 • Not Legitimate: 3 x 4 1 Inequality Symbols Don' t Have Same Sense 1 2 . 3 x 4 1 - 3 is NOT -1 1 2 Expressing Solutions to Three Part Inequalities • “Standard notation” - variable appears alone in the middle part of the three expressions being compared with two inequality symbols: 2 x 3 • “Graphical notation” – same as with two part inequalities: 2 3 ( ] • “Interval notation” – same as with two part inequalities: (2, 3] Solving Three Part Linear Inequalities • Solved exactly like two part linear inequalities except that solution is achieved when variable is isolated in the middle Example of Solving Three Part Linear Inequalities 3 1 x 4 1 2 1 3 x 2 1 2 6 x 4 2 2 x 2 Standard Notation Solution 2 2 [ ) Graphical Notation Solution [2, 2) Interval Notation Solution Homework Problems • Section: 2.8 • Page: 174 • Problems: Odd: 3 – 17, 21 – 25, 29 – 71 • MyMathLab Homework Assignment 2.8 for practice • MyMathLab Quiz 2.8 for grade Sets • A “set” is a collection of objects (elements) • In mathematics we often deal with sets whose elements are numbers • Sets of numbers can be expressed in a variety of ways: A 3, 6, 7, 11 Four specific numbersin the set B x | x 4 All numbers greater than 4 C 3, 8 All numbers - 3 and 8 D , 2 All numbers 2 Empty Set • A set that contains no elements is called the “empty set” • The two traditional ways of indicating the empty set are: Intersection of Sets • The intersection of two sets is a new set that contains only those elements that are found in both the first AND and second set • The intersection of sets M and N is indicated by M N • Given M , 2, 3, 4 and N 2, 4, 6, 8 1 M N 2, 4 Union of Sets • The union of two sets is a new set that contains all those elements that are found either in the first OR the second set • The intersection of sets M and N is indicated by M N • Given M , 2, 3, 4 and N 2, 4, 6, 8 1 M N 1, 2, 3, 4, 6, 8 Intersection and Union Examples • Given C 3, 8 and D , 2 • Find the intersection and then the union (it may help to first graph each set on a number line) • Find CD 3, 2 • Find CD , 8 Compound Inequalities • A compound inequality consists of two inequalities joined by the word “AND” or by the word “OR” • Examples: 2 x 3 3 AND x 2 5 2 x 5 OR x 3 5 Solving Compound Inequalities Involving “AND” • To solve a compound inequality that uses the connective word “AND” we solve each inequality separately and then intersect the solution sets • Example: 2 x 3 3 AND x 2 5 2 x 6 AND x 3 x 3 AND x 3 , 3 3, No number can be in both sets This means thereis no solution Solving Compound Inequalities Involving “OR” • To solve a compound inequality that uses the connective word “OR” we solve each inequality separately and then union the solution sets • Example: 2 x 5 OR x 3 5 x 3 OR x 8 x 3 OR x 8 3, 8, Always simplify 3, Solution Homework Problems • Section: 9.1 • Page: 626 • Problems: Odd: 7 – 61 • MyMathLab Homework Assignment 9.1 for practice • MyMathLab Quiz 9.1 for grade Definition of Absolute Value • “Absolute value” means “distance away from zero” on a number line • Distance is always positive or zero • Absolute value is indicated by placing vertical parallel bars on either side of a number or expression Examples: The distance away from zero of -3 is shown as: 3 3 The distance away from zero of 3 is shown as: 3 3 The distance away from zero of u is shown as: u Can' t be simplified , because value of " u" is unknown. However,its value is zero or positive. Absolute Value Equation • An equation that has a variable contained within absolute value symbols • Examples: | 2x – 3 | + 6 = 11 | x – 8 | – | 7x + 4 | = 0 | 3x | + 4 = 0 Solving Absolute Value Equations • Isolate one absolute value that contains an algebraic expression, | u | – If the other side is negative there is no solution (distance can’t be negative) – If the other side is zero, then write: • u = 0 and Solve – If the other side is “positive n”, then write: • u = n OR u = - n and Solve – If the other side is another absolute value expression, | v |, then write: • u = v OR u = - v and Solve Example of Solving Absolute Value Equation 2x 3 6 11 2x 3 5 2x 3 5 OR 2x 3 5 2x 8 2x 2 x4 x 1 Example of Solving Absolute Value Equation x 9 7x 4 0 x 9 7x 4 x 9 7x 4 OR x 9 7 x 4 13 6x x 9 7 x 4 13 8x 5 x 6 5 x 8 Example of Solving Absolute Value Equation 3x 6 2 3x 4 ? This says distanceis negative - NOT POSSIBLE! Equation has NO SOLUTION! Absolute Value Inequality • Looks like an absolute value equation EXCEPT that an equal sign is replaced by one of the inequality symbols • Examples: | 3x | – 6 > 0 | 2x – 1 | + 4 < 9 | 5x - 3 | < -7 Solving Absolute Value Inequalities 1. Isolate the absolute value on the left side to write the inequality in one of the forms: | u | < n or | u | > n 2a. If | u | < n, then write and solve one of these: u > -n AND u < n (Compound Inequality) -n < u < n Preferred! (Three part inequality) 2b. If | u | > n, then write and solve: u < -n OR u > n (Compound inequality) 3. Write answer in interval notation Example: Solve: | 3x | – 6 > 0 1. Isolate the absolute value on the left side to write the inequality in one of the forms: | u | < n or | u | > n 3x 6 2a. If | u | < n, then write: -n < u < n , and solve 2b. If | u | > n, then write: u < -n OR u > n , and solve Which formdoes this match? 2b Next? 3x 6 OR 3x 6 Example Continued 3x 6 OR 3x 6 Solve each inequality separately: x 2 OR x2 3. Answer in interval notation : (, 2) (2, ) Example: Solve: | 2x -1 | + 4 < 9 1. Isolate the absolute value on the left side to write the inequality in one of the forms: | u | < n or | u | > n 2x 1 5 2a. If | u | < n, then write: -n < u < n , and solve 2b. If | u | > n, then write: u < -n or u > n , and solve Which formdoes this match? 2a Next? 5 2 x 1 5 Example Continued 5 2 x 1 5 4 2x 6 2 x 3 How do we write the answer? 2, 3 Absolute Value Inequality with No Solution • How can you tell immediately that the following inequality has no solution? 5x 7 2 • It says that absolute value (or distance) is negative – contrary to the definition of absolute value • Absolute value inequalities of this form always have no solution: u n (where n represents a negative number) Does this have a solution? 2x 5 0 • At first glance, this is similar to the last example, because “ < 0 “ means negative, and: 2x 5 can't be less than a negative number! • However, notice the symbol is: • And it is possible that: 2x 5 0 • We have previously learned to solve this as: 2x 5 0 5 2x 5 Solution is : x x 5 2 2 Solve this: 4x 5 0 • Remember that absolute value of a number is always greater than or equal to zero, therefore the solution will be: • every real number except the one that makes this absolute value equal to zero (the inequality symbol says it must be greater than zero) • Another way of saying this is that: The only bad value of “x” is: 4x 5 0 5 x 4 5 5 • The solution, in interval notation is: , , 4 4 Homework Problems • Section: 9.2 • Page: 635 • Problems: Odd: 1, 5 – 31, 35 – 95 • MyMathLab Homework Assignment 9.2 for practice • MyMathLab Quiz 9.2 for grade