Algebra 1

1
Algebra 1

• Ch 1.4 Equations Inequalities

2
Objective

• Students will check solutions and solve equations

3
Vocabulary

• An equation is formed when an equal sign () is
  placed between two expressions creating a left
  and a right side of the equation
• An equation that contains one or more variables
  is called an open sentence
• When a variable in a single-variable equation is
  replaced by a number the resulting statement can
  be true or false
• If the statement is true, the number is a
  solution of an equation
• Substituting a number for a variable in an
  equation to see whether the resulting statement
  is true or false is called checking a possible
  solution

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Checking the Solution

• When checking a possible solution to an equation
  you will use the process that you learned in
  previous lessonsthat is
• Write the equation
• Substitute
• Simplify
• If the number substituted creates a true
  statement then it is a solution to the equation.
• If the substituted number creates a false
  statement then it is not a solution to the
  equation

5
Comments

• Be careful here!...In addition to using what you
  have learned in previous lessons we are taking
  this lesson one step further
• In this lesson you are being asked to analyze the
  end results and make a decisionis the end result
  true or false?
• This thought process is what Algebra is all
  aboutwe will show you how to solve problems in a
  logical sequential wayand then ask you to make
  meaning out of what you have done
• We will work with this concept throughout the
  course!

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Example 1

• Check whether the numbers 2, 3 4 are solutions
  to the equation 4x 2 10

4x 2 10
1. Write the equation
4(2) 2 10
2. Substitute 2 for x
3. Simplify
8 2 10
6 10
4. Analyze the result
5. Draw the conclusion
6 ? 10
This symbol means does not equal
Conclusion 2 is not a solution to the equation
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Example 2

• Check whether the numbers 2, 3 4 are solutions
  to the equation 4x 2 10

4x 2 10
1. Write the equation
4(3) 2 10
2. Substitute 3 for x
3. Simplify
12 2 10
10 10
4. Analyze the result
5. Draw the conclusion
10 10
Conclusion 3 is a solution to the equation
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Example 3

• Check whether the numbers 2, 3 4 are solutions
  to the equation 4x 2 10

4x 2 10
1. Write the equation
4(4) 2 10
2. Substitute 4 for x
3. Simplify
16 2 10
14 10
4. Analyze the result
5. Draw the conclusion
14 ? 10
Conclusion 4 is not a solution to the equation
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Comments

• Notice that in each of the examples the equal
  signs are lined up
• They are lined up that way so that it is easy to
  distinguish between the left and right side of
  the equations
• Students that are not organized have difficulty
  solving problems because they are not organized
  and get confused on which side of the equation to
  work.
• The ability to be organized and show step by step
  solutions to problems minimizes errors,
  demonstrates what you understand and begins to
  develop logical thinking processeswhich is what
  this course is all about!

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Real Life Application

• You probably already use the process that we just
  learned informally in your real life
• Suppose for your birthday you are given 50.00
  and you decide that you want to buy 2 video
  games.
• The cost of the games are 24.99 and 30.00 each.
  Mentally you have done a quick calculation and
  realize that the statement is false (25 30 55
  ? 50)therefore, you do not have enough money and
  cannot buy the 2 video games

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Inequalities

• Another type of open sentence is called an
  inequality.
• An inequality is formed when and inequality sign
  is placed between two expressions
• A solution to an inequality are numbers that
  produce a true statement when substituted for the
  variable in the inequality

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Inequality Symbols

• Listed below are the 4 inequality symbols and
  their meaning
• lt Less than
• Less than or equal to
• gt Greater than
• Greater than or equal to

Note We will be working with inequalities
throughout this courseand you are expected to
know the difference between equalities and
inequalities
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Equalities vs. Inequalities

• In an equality there is only one solution
• Example
• 2x 2 6
• We can use mental math to determine that the
  solution is 4.
• 4 is the only number that will make the above
  statement true

• In an inequality there are many solutions
• Example
• 2x 2 lt 6
• We can use mental math to determine that the
  solution is lt 4.
• Any number less than 4 will make this a true
  statement.
• The number 4 will not make this statement true,
  therefore it is not in the solution set

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Checking the Solution

• When checking a possible solution to an
  inequality you will use the process that you
  learned in previous lessonsthat is
• Write the inequality
• Substitute
• Simplify
• If the number substituted creates a true
  statement then it is a solution to the
  inequality.
• If the substituted number creates a false
  statement then it is not a solution to the
  inequality

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Example 4

• Decide if 4 is a solution to the inequality 2x
  1 lt 8

2x 1 lt 8
1. Write the inequality
2(4) 1 lt 8
2. Substitute 4 for x
8 1 lt 8
3. Simplify
7 lt 8
4. Analyze the result
5. Draw the conclusion
True
Conclusion 4 is a solution to the inequality
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Example 5

• Decide if 4 is a solution to the inequality x 4
  gt 9

x 4 gt 9
1. Write the inequality
4 4 gt 9
2. Substitute 4 for x
8 gt 9
3. Simplify
8 gt 9
4. Analyze the result
5. Draw the conclusion
False
Conclusion 4 is not a solution to the inequality
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Example 6

• Decide if 4 is a solution to the inequality x 3
  1

x 3 1
1. Write the inequality
4 3 1
2. Substitute 4 for x
1 1
3. Simplify
1 1
4. Analyze the result
5. Draw the conclusion
True
Conclusion 4 is a solution to the inequality
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Comments

• On the next couple of slides are some practice
  problemsThe answers are on the last slide
• Do the practice and then check your answersIf
  you do not get the same answer you must question
  what you didgo back and problem solve to find
  the error
• If you cannot find the error bring your work to
  me and I will help

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Your Turn Checking Equations

• Check whether the given number is a solution to
  the equation
• 3b 1 13 b4
• 6d 5 20 d 5
• 2y2 3 5 y 1
• p2 5 20 p 6
• m 4m 60 2m m 10

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Your Turn Checking Inequalities

• Check whether the given number is a solution to
  the inequality
• n 2 lt 6 n 3
• 4p 1 8 p 2
• y3 2 8 y 2
• 25 d 4 d 5
• d
• a(3a 2) gt 50 a 4

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Your Turn Word Problem

• You are playing a new computer game. Fore every
  eight screens you complete, you receive a bonus.
  You want to know how many bonuses you will
  receive after completing 96 screens. You write
  the equation 8x 96 to model the situation.
• What do 8, x and 96 represent?
• Solve the equation
• Check your solution

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Your Turn Solutions

• True
• False
• True
• False
• False
• True
• False
• True
• True
• True

• 11. a. 8 of screens for bonus, x bonus, 96
  number of screens played
• 11. b. x 12
• 11. c. 8(12) 96
• 96 96
• True

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Summary

• A key tool in making learning effective is being
  able to summarize what you learned in a lesson in
  your own words
• In this lesson we talked about checking solutions
  to equations and inequalitiesTherefore, in your
  own words summarize this lessonbe sure to
  include key concepts that the lesson covered as
  well as any points that are still not clear to
  you
• I will give you credit for doing this
  lessonplease see the next slide

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Credit

• I will add 25 points as an assignment grade for
  you working on this lesson
• To receive the full 25 points you must do the
  following
• Have your name, date and period as well a lesson
  number as a heading.
• Do each of the your turn problems showing all
  work
• Have a 1 paragraph summary of the lesson in your
  own words
• Please be advised I will not give any credit
  for work submitted
• Without a complete heading
• Without showing work for the your turn problems
• Without a summary in your own words