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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:

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Intermediate Algebra

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## Intermediate Algebra

Exam 3 Material

Inequalities and Absolute Value

### 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:

.

### 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”

### 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

### 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 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:

### 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.

### 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)

### 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:

.

### 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:

### SolvingThree Part Linear Inequalities

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

### 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:

### 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

### 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

### 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

### Compound Inequalities

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

• Examples:

### 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:

### 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:

### 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:

The distance away from zero of 3 is shown as:

The distance away from zero of u is shown as:

### 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 sideis another absolute value expression, | v |, then write:

• u = v OR u = - v and Solve

### 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

### 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)

### 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

### 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

### 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:

### 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:

### 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:

### 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