Algebra ReviewNumber Sense

• Whole Numbers: 0, 1, 2, 3, 4, …• Natural Numbers: 1, 2, 3, 4, … , • Integers : - -4, -3, -2, -1, 0, 1, 2, 3, 4, … , • Rational Numbers: any number expressed as a/b, where a

and b are integers and b cannot be 0. All whole numbers, natural numbers, and integers are also rational numbers.

• Irrational Numbers: any real number that cannot be expressed as a ratio of integers. This means its decimal part never ends and never repeats.

• Real Numbers: Real numbers is the group of numbers that include all rational numbers and all irrational numbers. If you think of a number line, the number line that you are most familiar with is often called the “Real Number Line” because every integer, rational number (fraction), and irrational number like have a place on the number line

• These subgroups of numbers are often represented visually using a Venn Diagram.

Algebra ReviewSolving Equations

Key Concepts

• Remember the overall goal for solving equations is to undo the implied operations (multiplication, division, subtraction, and addition) in an effort to isolate the variable term

• Variable Term – the use of a letter (or some symbol) to represent an unknown value.

• The word variable is used because it often refers to a number that can change or VARY

• When provided an equation you want to combine all like terms that show up on each side of the “=“ sign.

Example:

• Then if the variable shows up on both sides of the “=“ sign you must add or subtract one side from the other. What you do to one side you must do to both sides.

5 4 2 3 3 12x x x

7 4 3 9x x 3x3x

4 4 9x

• Now that the variable is visible on only one side we need to isolate it (get that term all by itself).

• Now that the variable term (4x) is all by itself we can work on getting the 4 turned into a 1. Remember that 4x actually means 4 times x. So to undo the multiplication we need to divide by the 4.

4 4 9x 4 4

4 13x

13

4x

• Once we isolate our variable we must determine how we want to write our solution. Should it stay an improper fraction? Should we re-write it as a mixed number? Should it be converted to a decimal?

• The answer to this question is often dictated by the type of problem, however we must make the distinction that leaving answers as reduced fractions or mixed numbers is the most appropriate way to leave an answer, often when a solution is rational (written as a fraction) its decimal form does not terminate and we have to round the answer, rounding can cause problems if that number needs to be used for other calculations. In this class, unless stated, assume that the answer needs to be left as a mixed number or reduced proper or improper fraction.

13

4x

• Alternate techniques for solving this equation.

• Instead of dividing both sides by 4, we can multiply by the coefficients multiplicative inverse (the number that you multiply 4 by to produce 1). In this case that number would be a fraction of ¼.

• Then we would do the same thing to the other side of the equation.

4 13x

Example 1

Example 2

Example 3

• Parentheses take precedence.

Example 4

Example 5

Homework assignment in MATHXL

Algebra ReviewSystems of Equations

• Systems of equations are used when you have an equation that has two variables. When working with equations with two variables it is impossible to determine the values of the two variables unless one is provided for you. However, sometimes one of the variables’ values is not provided. In cases such as this you will need another equation that relates the same two variables to one another.

• It is impossible to determine a numerical value for x if we don’t know what y is, and it is impossible to determine y if we don’t know x. Therefore we need another equation that uses the same two variables.

• Now that we have two equations with the same two variables, we want to make sure that the variables align on top of one another. X’s above X’s, and Y’s above Y’s. All other values on the other side of the = sign.

Now we use a technique called elimination in which we add straight down in all 3 columns in hopes that a variable

disappears (eliminates). Then the resulting equation will be of one variable.

6 36y

6 36y Once you have eliminated a variable you should have something that looks similar to the following. Now solve for that variable.

6y Remember we initially want to solve for x and y, now we know y, we can substitute it into any equation that involves y from this problem and solve for x.

Example 1

• Sometimes we need to alter one of (or both) equations so they eliminate. You may multiply one (or both) equations by any value you wish, as long as you multiply each term in the equation.

Example 2

Example 3

Example 4

Homework in MATHXL

Algebra ReviewRadical Expressions

• Radical Expression - an algebraic expression involving a square root, or an nth root.

• A square root is essentially a symbol that represents a particular number that when multiplied by itself it equates to the number under the radical. The is referred to as the radical, the number or expression underneath is the radicand. In this case a simple radical is an understood square root. Sometimes you will see a number out in front of the symbol, that number determines the root.

• 2 = square root• 3 = cube root• 4 = 4th root• 5 = 5th root

2

3

4

5

• In this course we will deal primarily with square roots with an occasional cube root. So when we ask what is the we are asking the question: “What number multiplied by itself will give us 8?”

• If we were given we are asking the question: “What number multiplied by itself 3 times will give us 8?”

2 8

3 8

• Unless the number under the radical is a perfect square, then the answer to the previous questions will always be an irrational number (a number whose decimal part never ends nor shows any patter of repeating). It is convenient to use a calculator to prove to ourselves that this concept is true. Take the following for examples.

Use your calculator to determine the numerical value of the following radical expressions

12

354

18

16

225

100

Radicals predominately leave you with an irrational number therefore converting the expression to a decimal requires rounding, thus we develop a technique to reduce radicals that allows for easier mathematical manipulation later on in more complicated processes. We usually want to avoid rounding as much as possible because it leads to slightly inaccurate answers.

• The technique for reducing radicals is called PRIME FACTORIZATION and you will know to complete this when you see the phrase, “leave answers in simplest radical form”

• PRIME FACTORIZATION – we will break down every radicand into two factors (numbers that multiply to give a product). We usually do this by thinking about what divides the radicand evenly. This will result in 2 numbers that are smaller than the original radicand. We will then factor both of those 2 smaller numbers and we will continue to do this and will only stop once we arrive at numbers that can only be factored/divided by the number 1 and itself (PRIME).

Example 1PRIME FACTORIZATION OF 28

Example 2PRIME FACTORIZATION OF 76

Example 3PRIME FACTORIZATION OF 324

• Now we will use the same process to help us determine the simplified form of

288288

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Operations with Radicals

• Adding radicals. We can only add like radicals. This means that if the radicands are identical we can add the coefficient out front, however, if the radicands are different combining cannot take place. Same with subtraction.

Example 10 : Simplify

4 6 9 3 12 6

Example 11

• These cannot be combined initially, however each radicand can be simplified further, and then hopefully with the reduction we can see like radicands.

Example 12

Multiplying Radicals

• Multiplying radicals is the easiest of radical rules to remember. Multiply the numbers that are on the outside by the numbers on the outside and they stay on the outside in the answer. Multiply the numbers on the inside by numbers on the inside and they stay on the inside in the answer. Then when all multiplying is done make sure the radicand is fully simplified/reduced.

Ex: 13

Ex. 14

Ex. 15

Ex. 16

Ex. 17

Algebra ReviewFactoring