1. Ch 1.4 – Equations & Inequalities
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

4. Checking the Solution
When checking a possible solution to an 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

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 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!

6. This symbol means does not equal
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
6 ≠ 10
5. Draw the conclusion
This symbol means does not equal
Conclusion: 2 is not a solution to the equation

7. 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
10 = 10
5. Draw the conclusion
Conclusion: 3 is a solution to the equation

8. 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
14 ≠ 10
5. Draw the conclusion
Conclusion: 4 is not a solution to the equation

9. 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!

10. 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 ( = 55 ≠ 50)…therefore, you do not have enough money and cannot buy the 2 video games

11. 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

12. 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

13. 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 < 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

14. Checking the Solution
When checking a possible solution to an inequality you will use the process that you learned in previous lessons…that 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

15. Example #4 2x – 1 < 8 2(4) – 1 < 8 8 – 1 < 8 7 < 8 True
Decide if 4 is a solution to the inequality 2x – 1 < 8
2x – 1 < 8
1. Write the inequality
2(4) – 1 < 8
2. Substitute 4 for x
8 – 1 < 8
3. Simplify
7 < 8
4. Analyze the result
True
5. Draw the conclusion
Conclusion: 4 is a solution to the inequality

16. Example #5 Decide if 4 is a solution to the inequality 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

17. Example #6 Decide if 4 is a solution to the inequality 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

18. 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…

19. Your Turn – Checking Equations
Check whether the given number is a solution to the equation
3b + 1 = 13 b=4
6d – 5 = 20 d = 5
2y2 + 3 = 5 y = 1
p2 – 5 = p = 6
m + 4m = 60 – 2m m = 10

20. Your Turn – Checking Inequalities
Check whether the given number is a solution to the inequality
n – 2 < 6 n = 3
4p – 1 ≥ 8 p = 2
y3 – 2 ≤ 8 y = 2
25 – d ≥ 4 d = 5 d
a(3a +2) > 50 a = 4

21. 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

22. 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…

23. 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…

24. Your Turn Solutions True False
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