Algebra I - Chapter 3 – Solving Inequalities

3-1 Inequalities and Their Graphs

An inequality is a mathematical sentence that uses an inequality symbol to compare the values of two expressions.

Example (Writing Inequalities): What inequality represents the verbal expression?

a) All real numbers x less than or equal to –7 b) 6 less than a number x is greater than 13

A solution of an inequality is any number that makes the inequality true. Substitute the values to determine if the value is a solution or not.

Example (Identifying Solutions by Evaluating): Is the number a solution of ?

a) –3 b) – 2 c) –1

You can use a graph to indicate all of the solutions of an inequality.

Example:

a) What is the graph of ? b) What is the graph of ?

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Example (Writing an Inequality From a Graph): What inequality represents the graph?

a) g b) h

c) j d) k

Phrase InequalityAt most 15At least 13No greater than 6No more than 8No less than -7More than 4Less than -2

Homework Exercises: Pages 168-170, (8-16 EVEN, 17-24, 29-39, 44-56 EVEN)

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3-2 Solving Inequalities Using Addition and Subtraction

Solve each equation.

x – 15 = -12 m + 6 = -4

Example (Using the Addition & Subtraction Property of Inequality):

Solve and graph the solutions to the following inequalities. Check your solutions.

a) b)

Homework Exercises: Pages 174-175, (14 - 28 EVEN, 34 - 44 EVEN, 54 - 64 EVEN)

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3-3 Solving Inequalities Using Multiplication or Division

******Multiplying or Dividing each side of an inequality by a negative number changes the meaning of the inequality. You need to reverse the inequality symbol to make the inequality true!!!!!!!!!!!!!!!!!!

Example:…begin with a true inequality and test it…

Example (Using the Multiplication & Division Property of Inequality):

Solve and graph the solutions to the following inequalities. Check your solutions.

a) b)

c) d)

Homework Exercises: Pages 181-183, (8-30 EVEN, 68-76 ALL)

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3-4 Solving Multi-Step Inequalities

Example: Solve for the variable. Show your steps. Check your solutions.

a) b) c)

Example (Geometry): In a community garden, you want to fence in a vegetable garden that is adjacent to your friend’s garden. You have at most 42 feet of fence. What are the possible lengths of your garden?

Example (Multi-Step Problems): Solve for the variable. Show your steps. Check your solutions.

a) b)

c) d)

Homework Exercises: Page 190, (10-42 EVEN)

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3-5 Working With Sets

Sets are the basis of mathematical language. You can write sets in different ways and form smaller sets of elements from a larger set. You can also describe the elements that are not in a given set.

Roster form is one way to write sets. Roster form lists the elements of a set within braces, { }. For example, you write the set containing the multiples of 2 as {2, 4, 6, 8, …}.

Set-builder notation is another way to write sets. It describes the properties an element must have to be included in the set. For example, you can write the set {2, 4, 6, 8, …}as {x│x is a multiple of 2}. This is read, “the set of all real numbers x, such that x is a multiple of 2.”

Example: How do you write “T is the set of natural numbers that are less than 6” in:

a) Roster form b) Set-builder notation

Example (Inequalities and Set-Builder Notation): Solve the inequalities below and write the solutions in set-builder notation.

a) b)

By definition, set A is a subset of set B if each element of A is also an element of B. For example, if B = and A = , then A is a subset of B, written A B.

The empty set or null set, is the set that contains no elements. (The empty set is a subset of every set) Use or { } to represent the empty set. **Do NOT use { }, as this set is not empty!

Example (Finding Subsets): What are all of the subsets of

a) {3, 4, 5}? b) {a, b}?

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When working with sets, you can call the largest set you are using the universal set, or universe. The complement of a set is the set of all elements in the universal set that are not in the set. The complement of set A is denoted by Aʹ.

In the Venn diagrams below, U represents the universal set. Notice that A U and Aʹ U.

Example: Universal set U = {months of the year}, and set A = {months with exactly 31 days}.

What is the complement of set A? Write your answer in roster form.

Homework Exercises: Pages 197-199, (10-28 EVEN, 30-33, 40-44 EVEN, 58-66 EVEN).

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3-6 Compound Inequalities

A compound inequality consists of two distinct inequalities joined by the work and or the word or.

Example (Writing and Graphing Compound Inequalities): Write and graph a compound inequality.

a) all real numbers that are greater than –2 b) all real numbers that are less than 0 or greater and less than 6 than or equal to 5

A solution to a compound inequality involving and is any number that makes both inequalities true. One way to solve a compound inequality is by separating it into two inequalities.Example: Solve the following inequalities using separation techniques. Graph the solutions.

a) b)

You can also solve an inequality like by working on all 3 parts of the inequality at the same time. You work to isolate the variable between the inequality symbols. Try this method below.

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Example: To earn a B in a certain high school’s algebra course, you need to achieve an unrounded test average between 84 and 86, inclusive. You score 86, 85, and 80 on the first 3 tests of a grading period. What possible scores can you earn on the fourth and final test to earn a B in the course?

A solution to a compound inequality involving or is any number that makes either inequality true. To solve a compound inequality involving or, you must solve separately the two inequalities that form the compound inequality.

Example: Solve the following inequalities. Graph your solutions.

a) or b) or

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You can use an inequality such as to describe a portion of the number line called an interval. You can also use interval notation to describe an interval on the number line. Interval notation includes the use of 3 special symbols: parentheses ( ), brackets [ ], and infinity ∞.

Example (Using Interval Notation):

a) What is the graph of ? How do you b) What is the graph of or ? How do write as an inequality? you write or in interval notation?

Homework Exercises: Pages 204-206, (12-40 EVEN, 56-61)

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3-7 Absolute Value Equations and Inequalities

You can solve absolute value equations and inequalities by first isolating the absolute value, and then writing an equivalent pair of linear equations or inequalities.

Example: Isolate the absolute value and solve the equations below. Check on paper and graphically.

a) b)

Example: Solve each absolute value equation. Check your solutions.

a) b)

c) d)

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When solving an absolute value inequality, I employ GorLand-Double-Switch techniques. Pay close attention to the type of inequality, and you will become very successful at solving these inequalities.

Key: G stands for “greater than” and L stands for “less than” (everything else is self-explanatory)

Example: Find all real solutions to the following. Graph your solutions.

a) b)

c) d)

Homework Exercises: Page 211, (18-46 EVEN)

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3-8 Unions and Intersections of Sets

The union of two or more sets is the set that contains all elements of the sets. The symbol for union is . An element is in the union if it belongs to at least one of the sets.

The intersection of two or more sets is the set of all elements that are common to every set. The symbol for intersection is . An element is in the intersection if it belongs to all of the sets.

Disjoint sets have no elements in common.

Below is a visual representation of union, intersection, and disjoint sets.

Example (Union): In your left pocket, you have a quarter, a paper clip, and a key. In your right pocket, you have a penny, a quarter, a marble, and a pen. Write a list for each pocket in roster form. Then write a set that represents the union of both sets.

Example (Intersection): Set X ={x│x is a natural number less than 19}; Y ={y│y is an odd integer}; and Z = {z│z is a multiple of 6}.

a) What is X Z ? b) What is Y Z ?

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Example (Using a Venn Diagram to Show Numbers of Elements): Of 500 commuters polled, some drive to work, some take public transportation, and some do both. 200 commuters drive to work and 125 use both types of transportation. How many commuters take public transportation?

Example: Of 30 students in student government, 20 are honors students and 9 are officers and honors students. All of the students are officers, honors students, or both. How many are officers but not honors students?

Example: What are the solutions of ? Write the solutions as either the intersection or union of two sets.

Homework Exercises: Pages 218-220, (10-32 EVEN, 50-58 EVEN)

Chapter Review Homework: Pages 223-226, (6-16, 20-26, 29-46, 49-56, 58-61)

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