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:

.

3315 xx inequalitypart Two

142

13 x inequalitypart Three

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

• 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

• 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

x : toequivalent is 23 x

x : toequivalent is 63 x

x : toequivalent is 82 x

5

2

4

3 Add

2-by Divide

3by Divide

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: 3315 xx

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:

• “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

• “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.

2x

]2

]2,(

SolvingTwo 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

3315 xx

9315 xx

62 x

3x

!inequality of sense reverse negative, aby dividingWhen

3

]

]3,(

SolutionNotation Standard

SolutionNotation Graphical

SolutionNotation Interval

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:

• Not Legitimate:

.

142

13 x

142

13 x

142

13 x

Sense Same Havet Don' Symbols Inequality

1- NOT is 3-

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:

32 x

( ]2 3

]3,2(

SolvingThree Part Linear Inequalities

• Solved exactly like two part linear inequalities except that solution is achieved when variable is isolated in the middle

Example of SolvingThree Part Linear Inequalities

142

13 x

122

13 x

246 x

22 x SolutionNotation Standard

2 2

[ ) SolutionNotation Graphical

SolutionNotation Interval)2,2[

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:

11,7,6,3A 4| xxB

8,3C

2,D

set in the numbers specificFour

4an greater th numbers All

8 and 3- numbers All

2 numbers All

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 and is indicated by

• Given and

NM M N

4,3,2,1M 8,6,4,2N

NM 4,2

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

NM M N

4,3,2,1M 8,6,4,2N

NM 8,6,4,3,2,1

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

8,3C 2,D

DC

DC

2,3

8,

Compound Inequalities

• A compound inequality consists of two inequalities joined by the word “AND” or by the word “OR”

• Examples:

52332 xANDx

5352 xORx

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: 52332 xANDx362 xANDx33 xANDx

,33, setsboth in becan number No

solution no is theremeans This

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:

,8,3 simplify Always

Solution

5352 xORx83 xORx83 xORx

,3

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 expressionExamples: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:

3

3

u

3

3

positive.or zero is valueits However, unknown. is u"" of valuebecause ,simplified bet Can'

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 SolvingAbsolute Value Equation

11632 x

532 x

532 x OR 532 x82 x 22 x

1x4x

Example of SolvingAbsolute Value Equation

0479 xx

479 xx

479 xx OR 479 xx

x613

x

6

13

479 xx

58 x

8

5x

Example of SolvingAbsolute Value Equation

263 x

43 x ?POSSIBLE! NOT - negative is distance says This

SOLUTION! NO hasEquation

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 (Three part inequality)

2b. If | u | > n, then write and solve:u < -n OR u > n (Compound inequality)

3. Write answer in interval notation

Preferred!

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

63 x

match? thisdoes formWhich 6363 xORx

2b Next?

Example Continued6363 xORx

:separately inequalityeach Solve

2x 2x

:notation intervalin Answer 3.

),2()2,(

OR

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

512 x

match? thisdoes formWhich 2a Next?

5125 x

Example Continued

5125 x

624 x

32 x

answer? the write wedo How

3,2

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:

275 x

)( numbernegativearepresentsnwherenu

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:

052 x

!numbernegativeathanlessbetcan'52 x

052 x

2

5

52

052

x

x

x

2

5 :isSolution x

Solve this:

• 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:

• The solution, in interval notation is:

054 x

054 x

4

5x

,

4

5

4

5,

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