Intermediate Algebra

1. Intermediate Algebra Exam 3 Material Inequalities and Absolute Value

2. Inequalities • An equation is a comparison that says two algebraic expressionsare 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: .

3. Inequalities • There are lots of different types ofinequalities, 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”

4. Properties of Inequalities • Adding or subtracting the same number to all parts of an inequality gives an equivalent inequalitywiththe samesense (direction) of the inequality symbol • Multiplying or dividing all parts of an inequality by the same POSITIVE number gives an equivalent inequalitywith the same sense (direction) of the inequality symbol • 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

5. 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

6. Two Part Linear Inequalities • A two part linear inequality is one that looks the same as a linear equationexcept 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:

7. Expressing Solutions to Two Part Inequalities • “Standard notation” - variable appears alone onleft side of inequality symbol, and a number appears alone onright side: • “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 squarebracket to indicate that a number is included • “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 alwaysused with a parenthesis.

8. SolvingTwo Part Linear Inequalities • Solve exactly like linear equationsEXCEPT: • 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)

9. Example of Solving Two Part Linear Inequalities

10. Three Part Linear Inequalities • Consist of three algebraic expressionscompared with two inequality symbols • Both inequality symbols MUSThave the same sense (point the same direction) AND must make a true statementwhen the middle expression is ignored • Good Example: • Not Legitimate: .

11. 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: • “Graphical notation” – same as with two part inequalities: • “Interval notation” – same as with two part inequalities:

12. SolvingThree Part Linear Inequalities • Solved exactly like two part linear inequalities except that solution is achieved when variable is isolatedin the middle

13. Example of SolvingThree Part Linear Inequalities

14. 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

15. 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:

16. Empty Set • A set that contains no elements is called the “empty set” • The two traditional ways of indicating the empty set are:

17. 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 and is indicated by • Given and

18. 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 and is indicated by • Given and

19. Intersection and Union Examples • Given and • Find the intersection and then the union (it may help to first graph each set on a number line) • Find • Find

20. Compound Inequalities • A compound inequality consists of two inequalities joined by the word “AND” or by the word “OR” • Examples:

21. 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:

22. 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:

23. Homework Problems • Section: 9.1 • Page: 626 • Problems: Odd: 7 – 61 • MyMathLab Homework Assignment 9.1 for practice • MyMathLab Quiz 9.1 for grade

24. 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: The distance away from zero of 3 is shown as: The distance away from zero of u is shown as:

25. 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

26. 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 sideis another absolute value expression, | v |, then write: • u = v OR u = - v and Solve

27. Example of SolvingAbsolute Value Equation

28. Example of SolvingAbsolute Value Equation

29. Example of SolvingAbsolute Value Equation

30. Absolute Value Inequality • Looks like an absolute value equationEXCEPT that an equal sign is replaced by one of the inequality symbols • Examples: | 3x | – 6 > 0 | 2x – 1 | + 4 < 9 | 5x - 3 | < -7

31. Solving Absolute Value Inequalities • 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 (Three part inequality) 2b. If | u | > n, then write and solve: u < -n OR u > n (Compound inequality) 3. Write answer in interval notation

32. 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 2a. If | u | < n, then write: -n < u < n , and solve 2b. If | u | > n, then write: u < -n OR u > n , and solve

33. Example Continued

34. 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 2a. If | u | < n, then write: -n < u < n , and solve 2b. If | u | > n, then write: u < -n or u > n , and solve

35. Example Continued

36. Absolute Value Inequalitywith No Solution • How can you tell immediately that the following inequality has no solution? • 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:

37. Does this have a solution? • At first glance, this is similar to the last example, because “ < 0 “ means negative, and: • However, notice the symbol is: • And it is possible that: • We have previously learned to solve this as:

38. Solve this: • Remember that absolute value of a number is always greater than or equal to zero, therefore the solution will be: • every real numberexceptthe 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: • The solution, in interval notation is:

39. 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