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 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)


Example of Solving Two 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

  • “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


Example of SolvingThree Part Linear Inequalities


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


Example of SolvingAbsolute Value Equation


Example of SolvingAbsolute Value Equation


Example of SolvingAbsolute Value Equation


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)

    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

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 Continued


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


Example Continued


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


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