Chapter 1-3 : Calculator Key-In

Does your calculator follow the steps for simplifying an expression stated on previous material?

Experiment with your calculator by entering the following example exactly as it appears here :

8 + 3 x 4 = ...

If your calculator displays the answer 20, it followed the order of operations you learned : multiplication before addition. Your calculator has an algebraic operating system. The answer 20 is correct.

If your calculator displays the answer 44, it performed the addition and the multiplication in the order in which you pressed the keys. One way to get the correct answer on your calculator is to multiply 3 and 4 first and then add 8, just as you would if you were using pencil and paper.

Exercises

Use your calculator to simplify each expression.

By : Mr. Danielz

Chapter 1-3. Introduction to Algebra : "EQUATIONS"

Objective :  To find solution sets of equations over a given domain.

An equation is formed by placing an equals sign between two numerical or variable expressions, called the sides of the equation.

Sentences containing variables (like the equations 5x - 1 = 9 and y + 2 = 2 + y) are called open sentences. The given set of numbers that a variable may represent is called the domain of the variable.

A variable in an equation can be replaced by any of the numbers in its domain. The resulting equation may be either true or false.

You may use braces { } to show a set of numbers. A short way to write "the set whose member are 1, 2, and 3" is {1, 2, 3}.

Example 1

The domain of x is {1, 2, 3}.

Is the equation 5x - 1 = 9 true when x = 1? when x = 2? when x = 3?

In example 1, when x replaced by 2, the resulting equation is true. Any value of a variable that turns an open sentence into a true statement is a solution, or root, of the sentence and is said to satisfy the sentence.

The set of all solutions of an open sentence is called the solution set of the sentence. Finding the solution set is called solving the sentence. In Example 1, there is only one solution. For the equation 5x - 1 = 9 you may say either "The solution is 2", or "The solution set is {2}".

Some equations have more than one solution, and some equations have no solutions. The sentence y + 2 = 2 + y is true no matter what number is substituted for y. Therefore the solution set is the set of all number. If you are asked to solve the equation over the domain {0, 1, 2, 3}, you state that the solution set is the domain itself, {0, 1, 2, 3}.

Here is another way to show that the domain of a variable y is {0, 1, 2, 3} :

(Read "y belongs to the set whose members are 0, 1, 2, and 3.")

Example 2

Example 3

Solve over the domain {6, 8, 12} :

Five more than twice a number is 29. What is the number ?

By : Mr. Danielz

Chapter 1-2. Introduction to Algebra : "GROUPING SYMBOLS"

Grouping Symbols

Objective : To simplify expressions with and without grouping symbols.

Parentheses have been used to show you how to group the numerals in an expression. Different groupings may produce different numbers.

A grouping symbol is a device, such as a pair of parentheses, used to enclose an expression that should be simplified first. Multiplication symbols are often left out of expressions with grouping symbols.

Example 1      Simplify :   a. 6(5 - 3)          b. 6(5) - 3

Solution

a. 6(5 - 3) stands for 6 x (5 - 3).

The parentheses tell you to simplify 5 - 3 first. Then multiply by 6.

6(5 - 3) = 6(2) = 12   Answer

b. 6(5) stands for 6 x 5.

6(5) - 3 = 30 - 3 = 27   Answer

In example 1, note that 6(5) stands for 6 x 5. Other ways to write this product using parentheses are (6)5 and (6)(5).

Example 2

Throughout your work in algebra, you will use these symbols :

Grouping Symbols

If an expression contains more than one grouping symbols, first simplify the expression in the innermost grouping symbol. Then work toward the outermost grouping symbol until the simplest expression is found.

Example 3

When there are no grouping symbols, simplify in the following order :

1. Do all multiplications and divisions in order from left to right.

2. Do all additions and substractions in order from left to right.

Example 4

Example 5

You may wish to use a calculator to evaluate some expressions.

By : Mr. Danielz

Chapter 1-1. Introduction to Algebra : "APPLICATION OF VARIABLES"

Energy Consumption

Power is associated with the flow of electricity in a circuit. Your electric company determines your monthly electric bill based on how much electricity you have used. Electrical power is measured in watts (W).

When p watts are used for t hours, the amount of energy measured in watt-hours is represented by the expression p.t ("p dot t"). The electric meter for your home measures the amount of electricity you use in units called kilowatt-hours (kW.h).

A kilowatt is 1000 watts.

To find the number of kilowatt-hours an appliance uses, use the expression p.t to determine the number of watt-hours used and then divide by 1000. You can get an idea of what 1 kW.h of electricity is by thinking of a 100-watt light bulb. To use 1 kW.h of electricity, you need to burn the 100-watt light bulb for 10 hours.

Example

An air conditioner uses 1330 watts for 6 hours. How many kilowatt-hours does it use?

Exercises

1. An iron using 1008 watts is plugged in for 2 hours. How many kilowatt-hours are used?

2. A clothes dryer uses 4856 watts. How many kilowatt-hours are used if it runs for 10 hours?

By : Mr. Danielz

Chapter 1-1. Introduction to Algebra : "VARIABLES" (Advanced)

continued from previous material

Replacing a numerical expression by the simplest name for its value is called simplifying the expression. When you simplify a numerical expression, you use the following principle.

Substitution Principle

An expression may be replaced by another expression that has the same value.

Example 1   Simplify each expression.

Replacing each variable in a variable expression by a given value and simplifying the result is called evaluating the expression or finding the value of the expression.

Example 2

In Examples 1 and 2 the parentheses show how the variables and numbers in the expression are to be grouped. Notice that expression within parentheses should be simplified first.

Example 3

Chapter 1-1. Introduction to Algebra : "VARIABLES"

VARIABLES

Objective : To simplify numerical expressions and evaluate algebraic expressions.

When you go to an ocean beach, you may find small shops that rent recreational equipment, such as scuba gear, surfboards, and snorkeling gear. Suppose the rental charge for snorkeling gear is $4.50 per hour. The amount you'll pay for using the gear depends on the amount of time you have it.

The rental charge follows this pattern :

Rental charge = $4.50 x number of hours

                         = $4.50 x h

The letter h stands for the hours shown in the table : 1, 2, 3, or 4. Also, h can stand for other hours not in the table. We call h a variable.

A variable is a symbol used to represent one or more numbers. The numbers are called values of the variable. An expression that contains a variable, such as the expression 4.50 x h , is called a variable expression. An expression, such as 4.50 x 4, that names a particular number is called a numerical expression, or numeral.

Another way to indicate multiplication is to use a raised dot, for example, 4.5 . 4.

In Algebra, products that contain a variable are usually written without multiplication sign because it looks to much like the letter x, which is often used as a variable.

The number named by a numerical expression is called the value of the expression. Since the expression 4 + 2 and 6 name the same number, they have the same value. To show that these expression have the same value, you use the equals sign, =.

You write               4 + 2 = 6

and say "Four plus two equals (or is equals to or is) six." The simplest, or most common, name for the number six is 6.

to show that the expression 4 + 2 and 5 do not have the same value.