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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

Intermediate Algebra

Exam 3 Material

Inequalities and Absolute Value

inequalities
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:

.

inequalities1
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
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
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
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
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.
solving two part linear inequalities
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
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
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:
solving three part linear 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
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
slide15
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
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
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
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
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
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
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
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 problems1
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
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
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
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
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
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

example solve 3x 6 0
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
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 inequality with no solution
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
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
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 problems2
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
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