Algebra 1 Ch 1.4 – Equations & Inequalities. Objective Students will check solutions and solve equations

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

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

equation you will use the process that you learned in previous lessons…that is

1. Write the equation2. Substitute3. 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

Conclusion: 2 is not a solution to the equation

6 = 10 4. Analyze the result

5. Draw the conclusion6 ≠ 10This symbol means does not equal

Example #2


4x – 2 = 10

4(3) – 2 = 10

12 – 2 = 10



3. Simplify

Conclusion: 3 is a solution to the equation


5. Draw the conclusion10 = 10

Example #3


4x – 2 = 10

4(4) – 2 = 10

16 – 2 = 10



3. Simplify

Conclusion: 4 is not a solution to the equation


5. Draw the conclusion14 ≠ 10

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 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

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 inequality2. Substitute3. 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

1. Write the inequality


3. Simplify

Conclusion: 4 is a solution to the inequality

7 < 8 4. Analyze the result

5. Draw the conclusionTrue

Example #5

Decide if 4 is a solution to the inequality x + 4 > 9

x + 4 > 9

4 + 4 > 9

8 > 9



3. Simplify

Conclusion: 4 is not a solution to the inequality

8 > 9 4. Analyze the result

5. Draw the conclusionFalse

Example #6

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

x – 3 ≥ 1

4 – 3 ≥ 1

1 ≥ 1



3. Simplify

Conclusion: 4 is a solution to the inequality

1 ≥ 1 4. Analyze the result

5. Draw the conclusionTrue

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 b=4

2. 6d – 5 = 20 d = 5

3. 2y2 + 3 = 5 y = 1

4. p2 – 5 = 20 p = 6

5. m + 4m = 60 – 2m m = 10

Your Turn – Checking Inequalities

Check whether the given number is a solution to the inequality

6. n – 2 < 6 n = 37. 4p – 1 ≥ 8 p = 28. y3 – 2 ≤ 8 y = 29. 25 – d ≥ 4 d = 5

d10. a(3a +2) > 50 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 equationc. 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: 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…

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

Documents

Algebra 1 Ch 1.4 – Equations & Inequalities. Objective Students will check solutions and solve equations