PRE-ALGEBRA

Pre-Algebra is the first math course in high school and will guide you through among other things integers, one-step equations, inequalities and equations, graphs and functions, percent, probabilities.

This Pre Algebra math course is divided into 3 chapters and each chapter is divided into several lessons. Each lesson includes theory and examples. I hope that you will enjoy studying Pre Algebra.

 

 1. INDRODUCING ALGEBRA

Ø Operations in the correct order :

 

When you are faced with a  mathematical expression that has several operations or parentheses, the solution may be affected by the order in which you tackle the operations.

 

For example, take the expression:

 

   4 . 7 - 2

 

If we do the Multiplication first, we arrive at the following answer :

 

   28 - 2 = 26

 

If instead we begin by subtracting , we get :

 

   4 . 5 = 20

 

In order to avoid confusion and to ensure that everyone always arrive at the same result, mathematicians established a standard order of operations for calculations that involve more than one arithmetic operations. Arithmetic operations should always be carried out in the following order:

 

  a) Simplify the expression inside parentheses ( ), brackets [ ], braces { } and 

       fraction bars.

  b) Evaluate all powers.

  c) Do all multiplications and divisions from left to right.

  d) Do all additions and subtractions from left to right.

 

 

 

Example :

 

Suppose you want to figure out how many hours  a person works  in 2 days assuming that they work 4 hours before lunch and 3 hours after lunch each day.

 

First, work out how many hours a person works each day:

 

    4 + 3 = 7

 

and then multiply that by the number of days the person worked:

 

    7 . 2 = 14

 

If we were to write this example as one expression, we would need to use parentheses to make sure that people calculate the addition first:

 

    (4 + 3) . 2 = 14

 

 Simplify the following expression :

 

     2 . [(8 - 3) + 6 . (2)] - 3

     2 . [ 5 + 12 ] - 3

     2 . 17 - 3

     34 - 3

     31

 

Ø Evaluate Expressions :

 

       A variable  is a letter, for example x, y or z , that represents an unspecified number.

 

  6 + x = 12

 

To evaluate an algebraic expression, you have to substitute a number for each variable and perform the arithmetic operations. In the example above, the variable x is equal to 6 since 6 + 6 = 12.

 

If we  know the value of our variables, we can replace the variables with their values and then evaluate the expression.

 

Example :

 

Calculate the following expression for x = 3 and z = 2

 

      6z + 4x = ?

 

 

Solution : Replace x with 3 and z with 2 to evaluate the expression.

 

     6 . 2 + 4 . 3 = ?

    12 +  12  =  24

 

     Evaluate the expression for x = 2, y = 5 and z = 4

          

           4x + ( 7 - z ) - 6y

           4 . 2 + ( 7 - 4 ) - 6 . 5

           8 + 3 - 30

           11 - 30

          -19

 

Ø Identifying  properties :

 

In this section, you will learn how to identify the properties of multiplication and addition and how you can use  identification to help solve mathematical problems.

 

To solve rather hard problems without using a calculator you have to identify all expressions that have the same operation.

 

Example :

 

     58 + 69 + 91 = ?

 

In this example, you can add in any order you prefer. The sum of the expression will not change if you prefer to add the numbers in a different order

 

    91 + 58 + 69 = 218

    58 + 69 + 91 = 218

 

The same is true for multiplication :

 

    5 . 4 . 30 = 600

    4 . 30 . 4 = 600

 

  Show that the following equation holds true

 

   3 . a . 4 = a . 4 . 3

   12 . a    =  a . 12

   12a = 12a

 

Ø Equations with variables:

 

 In this section, you will learn how to solve the equations that contain unknown variables. You will learn how to solve equations mentally by using the multiplication table and you will also learn how to identify a solution to an equation with given numbers as well as by using inverse operations.

 

You can solve an easy equation in your head  by using the multiplication table.

 

Example:

 

     8 . x = 64

 

Which number should you multiply 8 by to get a product of 64? By using the multiplication table, we know that the number is 8.

 

     8 . 8 = 64

 

When we solve an equation, we figure out what value of x ( or any other variable) makes the statement true ( satisfies the equation).

Example:

Which of the following numbers is a solution to the equation? x = 2, 7,  or 8?

      14 − x = 7

Here you are given the numbers 2, 7 and 8. One of these numbers will satisfy the equation. If you don't know the solution right away, you can investigate which of the given numbers gives results in the correct answer by plugging in the different values of x.

     x = 2 ⇒ 14 − 2 = 12             Wrong

     x = 7 ⇒ 14 − 7 = 7               Correct

     x = 8 ⇒ 14 − 8 = 6                Wrong

Answer: x = 7

You have already solved equations where the solutions are quite easy to see, by using mental math or patterns. Most equations are harder to solve and you have to simplify the equation before you can see the solution. One way to do this is to use inverse operations.

An operation is, for example, addition, multiplication, division and subtraction. An inverse operation is an operation that reverses the effect of another operation. Addition and subtraction are inverses of each other, just like division and multiplication are inverses.

 

Ø Coordinate system and ordered pairs

A coordinate system is a two-dimensional number line, for example, two perpendicular number lines or axes.

This is a typical coordinate system:

                             

The horizontal axis is called the x-axis and the vertical axis is called the y-axisThe center of the coordinate system (where the lines intersect) is called the origin. The axes intersect when both x and y are zero. The coordinates of the origin are (0, 0).

An ordered pair contains the coordinates of one point in the coordinate system. A point is named by its ordered pair of the form of (x, y). The first number corresponds to the x-coordinate and the second to the y-coordinate.

To graph a point, you draw a dot at the coordinates that corresponds to the ordered pair. It's always a good idea to start at the origin. The x-coordinate tells you how many steps you have to take to the right (positive) or left (negative) on the x-axis. And the y-coordinate tells you have many steps to move up (positive) or down (negative) on the y-axis.

Example :

                                         

The ordered pair (3, 4) is found in the coordinate system when you move 3 steps to the right on the x-axis and 4 steps upwards on the y-axis.

The ordered pair (-7, 1) is found in the coordinate system when you move 7 steps to the left on the x-axis and 1 step upwards on the y-axis.

To find out the coordinates of a point in the coordinate system you do the opposite. Begin at the point and follow a vertical line either up or down to the x-axis. There is your x-coordinate. And then do the same but following a horizontal line to find the y-coordinate.

Ø Inequalities

Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =

                          x = y                       x is equal to y

Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

                          x > y                       x is greater than y

                          x ≥ y                       x is greater than or equal to y

                          x < y                       x is less than y

                          x ≤ y                       x is less than or equal to y

An equation or an inequality that contains at least one variable is called an open sentence.

When you substitute a number for the variable in an open sentence, the resulting statement is either true or false. If the statement is true, the number is a solution to the equation or inequality.

Example :

Is 3 a solution to this equation?

                      5 x + 14 = 24

Substitute 3 for x

                      5⋅3 + 14 = 29 ≠ 24             FALSE!

Since 29 is not equal to 24, 3 is not a solution to the equation.

 

Example:

Is the following inequality true or false?

·        x − 4 > 12  , x = 13

             13 − 4 > 12       → false

·        y + 5 < 13 , y = 6

              6 + 5 < 13        → true

 

2. EXPLORE AND UNDERSTAND INTEGERS

Ø Absolute Value

 

In this section you'll learn how to the find the absolute value of integers.

4 - 0 = 4
4 - 1 = 3
4 - 2 = 2
4 - 3 = 1
4 - 4 = 0
4 - 5 = -1

In this pattern you can see that 4 - 5 is equal to a negative number. A negative number is a number that is less than zero (in this case -1). A negative number is always less than zero, 0. We can study this in a diagram by using two examples:

  0 - 4 = -4 and -1 - 3 = -4

                     

                                                 This kind of diagram is called a number line. There are some things that you need to observe when you draw and/or use a number line. Zero, is always in the middle and separates negative and positive numbers. On the left side of zero, you'll find numbers that are less than zero, the negative numbers. On the right side of zero, you'll find numbers that are greater than zero, the positive numbers. The absolute value is the same as the distance from zero of a specific number.

                                 

On this number line you can see that 3 and -3 are on the opposite sides of zero. Since they are the same distance from zero, though in opposite directions, in mathematics they have the same absolute value, in this case 3. The notation for absolute value is to surround the number by straight lines as in the examples below.

 

Example:

 

   Simplify the following expression

           

   |3| + |−3| = ?

    

    3 + 3 = 6

 

Ø Adding and subtracting integers

You already know how to add 3 + 4 and so on. But there are many ways to add integers. One way to add integers is by using a number line.

Example :

        −4 + (−3) = ?

                  

We always start at zero. Our first number is negative four (-4) so we move 4 units to the left. We then have plus negative three (-3) which is the same as subtracting 3 so we move 3 more units to the left. This gives us the value of negative seven, (-7).

       −4 + (−3) = ?

       −4 − 3 = −7

We do the same thing if we have a positive integer, but instead we move to the right.

       −4 + 3 = ?

       −4 + 3 = −1

                        

You can also add integers and variables.

Example:

           13 x + (−2) x = ?

           13 x − 2 x =  (13−2) x = 11x

When subtracting something from something we wish to find out the difference between the two numbers. When you subtract a negative number from any number the difference is even bigger. The distance from the seabed at a depth of 150ft and an airplane flying at 3000ft altitude at sea level is

         3000 − (−150) = 3000 + 150 = 3150 ft

Thus when we subtract negative numbers, we get:

        4 − (−3) = 4 + 3 = 7

Subtracting −3 is the same as adding 3.

If we have a plus sign before the parentheses then we will not change the signs within the parentheses

If we have a minus sign before the parentheses then we the signs within the parentheses will change.

Two negatives make one positive!

Ø Multiplying and Dividing integers

You also have to pay attention to the signs when you multiply and divide. There are two simple rules to remember:

When you multiply a negative number by a positive number then the product is always negative.

When you multiply two negative numbers or two positive numbers then the product is always positive.

This is similar to the rule for adding and subtracting: two minus signs become a plus, while a plus and a minus become a minus. In multiplication and division, however, you calculate the result as if there were no minus signs and then look at the signs to determine whether your result is positive or negative. Two quick multiplication examples:

               3 ⋅ (−4) = −12

3 times 4 equals 12. Since there is one positive and one negative number, the product is negative 12.

               (−3) ⋅ (−4) = 12

Now we have two negative numbers, so the result is positive.

Turning to division, you may recall that you can confirm the answer you get by multiplying the quotient by the denominator. If you answer is correct then the product of these two numbers should be the same as the numerator. For example,

              12 / 3 = 4

In order to check whether 4 is the correct answer, we multiply 3 (the denominator) by 4 (the quotient):

              3 ⋅ 4 = 12

What happens when you divide two negative numbers? For example,

             (−12) / (−3) = ?

For the denominator (-3) to become the numerator (-12), you would have to multiply it by 4, therefore the quotient is 4.

So, the quotient of a negative and a positive number is negative and, correspondingly, the quotient of a positive and a negative number is also negative. We can conclude that:

When you divide a negative number by a positive number then the quotient is negative.

When you divide a positive number by a negative number then the quotient is also negative.

When you divide two negative numbers then the quotient is positive.

The same rules hold true for multiplication.

 

 

3. INEQUALITIES AND ONE STEP EQUATION

Ø Different ways to solve equation

We have 4 ways of solving one-step equations: Adding, Substracting, multiplication and division. If we add the same number to both sides of an equation, both sides will remain equal.

Example :

                x = y

                x + z = y + z

If we subtract the same number from both sides of an equation, both sides will remain equal.

Example :

               x = y

               x − z = y − z

If we divide both sides of an equation by the same number, both sides will remain equal.

Example:

              x = y

              x ÷ z = y ÷ z

If we multiply both sides of an equation  by the same number, both sides will remain equal.

             x = y

             x ⋅ z = y ⋅ z

Ø Calculating the area and perimeter

The perimeter is the length of the outline of a shape. To find the perimeter of a rectangle or square you have to add the lengths of all the four sides. x is in this case the length of the rectangle while y is the width of the rectangle.

The perimeter, P, is:

             P = x + x + y + y

             P = 2x + 2y

             P = 2 (x+y)

Example:

Find the perimeter of this rectangle:

           

                 P = 7 + 7 + 4 + 4

                 P = 2 ⋅ 7 + 2 ⋅ 4

                 P = 2 ⋅ (7+4)

                 P = 2⋅ 11

                 P = 22 in

The area is measurement of the surface of a shape. To find the area of a rectangle or a square you need to multiply the length and the width of a rectangle or a square       

Area, A, is x times y.

                    A = x ⋅ y

 

Example :

Find the area of this square.

           

                  A = x ⋅ y

                  A = 5 ⋅ 5

                  A = 25 in2

There are different units for perimeter and area The perimeter has the same units as the length of the sides of rectangle or square whereas the area's unit is squared.

Ø Solving Inequalities

When we add or subtract the same number on both sides of the truth of the inequality doesn't change.

This holds true for all numbers:

                   x > y → x < y →

                   x + z > y + z → x + z < y + z  →

                   x − z > y − z       x − z < y − z

Example:

                  x + 3 > 9

                  x + 3 − 3 > 9 − 3

                 x > 6

It is a little bit trickier when it comes to division and multiplication

When we multiply or divide an inequality by a positive integer, the truth of the inequality doesn't change.

                x > y →

                x ⋅ z > y ⋅ z →

                x/z > y/z

                If  z>0

When we multiply or divide an inequality by a negative integer, the sign of the inequality will be reversed (changed).

                x > y →

                x ⋅ z < y ⋅ z →

                x /z < y/z

                If  z<0

Example :

               x / − 2 ≥ 3

               x/ − 2 ⋅ −2 ≥ 3 ⋅ −2

               x ≤ −6

Ø Understanding inequalities and equations

To expand our knowledge in equations solving, we need to solve some verbal problems. With verbal problems we mean problems that we need to translate into an equation by ourselves.

Example:

Leonard earns $60 a day. After a few days Leonard has earned $240. How many days has Leonard worked? We denote the number of days he has worked as x.

             $60 x = $240

             $60 x 60 = $24060

             x=4  To earn $240 Leonard has to work 4 days.