Transcript equation

Algebra 1
Ch 1.4 – Equations & Inequalities
Objective
 Students will check solutions and solve
equations
Vocabulary
 An equation is formed when an equal sign (=) is
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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
Checking the Solution
 When checking a possible solution to an
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equation you will use the process that you
learned in previous lessons…that 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
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 decision…is the
end result true or false?
 This thought process is what Algebra is all
about…we will show you how to solve
problems in a logical sequential way…and then
ask you to make meaning out of what you have
done…
 We will work with this concept throughout the
course!
Example #1
 Check whether the numbers 2, 3 & 4 are solutions to the
equation 4x – 2 = 10
4x – 2 = 10
4(2) – 2 = 10
8 – 2 = 10
1. Write the equation
2. Substitute 2 for x
3. Simplify
6 = 10
4. Analyze the result
6 ≠ 10
5. Draw the conclusion
This symbol means does not equal
Conclusion: 2 is not a solution to the equation
Example #2
 Check whether the numbers 2, 3 & 4 are solutions to the
equation 4x – 2 = 10
4x – 2 = 10
4(3) – 2 = 10
12 – 2 = 10
1. Write the equation
2. Substitute 3 for x
3. Simplify
10 = 10
4. Analyze the result
10 = 10
5. Draw the conclusion
Conclusion: 3 is a solution to the equation
Example #3
 Check whether the numbers 2, 3 & 4 are solutions to the
equation 4x – 2 = 10
4x – 2 = 10
4(4) – 2 = 10
16 – 2 = 10
1. Write the equation
2. Substitute 4 for x
3. Simplify
14 = 10
4. Analyze the result
14 ≠ 10
5. Draw the conclusion
Conclusion: 4 is not a solution to the equation
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 processes…which is what this course is all
about!
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
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
Inequality Symbols
 Listed below are the 4 inequality symbols and
their meaning
<
≤
>
≥
Less than
Less than or equal to
Greater than
Greater than or equal to
Note: We will be working with inequalities
throughout this course…and you are expected to
know the difference between equalities and
inequalities
Equalities vs. Inequalities
 In an equality there is only
 In an inequality there are
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
many solutions
Example:
2x – 2 < 6
 We can use mental math to
determine that the solution is
< 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
Checking the Solution
 When checking a possible solution to an
inequality you will use the process that you
learned in previous lessons…that is
1. Write the inequality
2. Substitute
3. 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
Example #4
 Decide if 4 is a solution to the inequality 2x – 1 < 8
2x – 1 < 8
2(4) – 1 < 8
8–1<8
7<8
True
1. Write the inequality
2. Substitute 4 for x
3. Simplify
4. Analyze the result
5. Draw the conclusion
Conclusion: 4 is a solution to the inequality
Example #5
 Decide if 4 is a solution to the inequality x + 4 > 9
x+4>9
1. Write the inequality
4+4>9
2. Substitute 4 for x
8>9
3. Simplify
8>9
4. Analyze the result
False
5. Draw the conclusion
Conclusion: 4 is not a solution to the inequality
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
True
5. Draw the conclusion
Conclusion: 4 is a solution to the inequality
Comments
 On the next couple of slides are some
practice problems…The answers are on the
last slide…
 Do the practice and then check your
answers…If you do not get the same answer
you must question what you did…go back
and problem solve to find the error…
 If you cannot find the error bring your work to
me and I will help…
Your Turn – Checking Equations
 Check whether the given number is a
solution to the equation
1. 3b + 1 = 13
2. 6d – 5 = 20
3. 2y2 + 3 = 5
4. p2 – 5 = 20
5. m + 4m = 60 – 2m
b=4
d=5
y=1
p=6
m = 10
Your Turn – Checking Inequalities
 Check whether the given number is a
solution to the inequality
n–2<6
4p – 1 ≥ 8
y3 – 2 ≤ 8
25 – d ≥ 4
d
10. a(3a +2) > 50
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n=3
p=2
y=2
d=5
a=4
Your Turn – Word Problem
11. 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.
a. What do 8, x and 96 represent?
b. Solve the equation
c. Check your solution
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 inequalities…Therefore, in your own
words summarize this lesson…be 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 lesson…please see
the next slide…
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:
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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:
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Without a complete heading
Without showing work for the your turn problems
Without a summary in your own words…
Your Turn Solutions
1. True
2. False
3. True
4. False
5. False
6. True
7. False
8. True
9. True
10. 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