Algebra Notes

Order of Operations Parentheses Exponents Multiplication Division Addition Subtraction

Variable Using arithmetic you might write something ___ + 4 + 37 = 45. You need a placeholder for this space. A placeholder in algebra is called a variable because the value can change

Algebraic Expression An Algebraic Expression is a combination of variables, number and at least one operation

Properties Distributive Property: is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Associative Properties: The way in which factors and addends are grouped does not change the sum or product. Identity Properties: The sum or product of an addend or factor and zero is zero Cumulative Properties: The order in which numbers are multiplies does not change the product.

Inverse Operations Using the opposite operation listed in the problem to undo the operation and find the solution. In Algebra this is also used to cancel out like terms

Use an Equation Mizzou beat Illinois in Football this weekend buy 10 points, Mizzou scored 23 points. 36 total points were scored by both teams combined. Write an equation for how many points Illinois scored. 23 + Y = 36 -23 -23 Y = 13 Can you think of your own equation?

Inequalities < and > are signs for inequalities. Inequalities are things that are not equal.

Integers and Absolute Value Absolute Value is the distance a number is from zero A coordinate is a point on the number line Integers can be written …-2, -1 0 1, 2, … … means continues without end

Adding Integers To add integers with different signs, subtract their absolute values. Give the results the same sign as the integers with the greater absolute value.

Subtracting Integers Opposites: Positive Integer paired with a negative Integer Additive Inverse: A number and it’s opposite are called additive inverses of each other Question: What do you notice about the sum of a number and it’s additive inverse?

Additive Inverse Continued… Additive Inverse can be used to define subtraction for integers Compare the following addition and subtraction sentences. Subtraction Addition 8-3=5 8+(-3)=5

Sample Problems 6-11 is the same as 6+(-11) 6+(-11) = -4-(-7)= 7-4 is the same as 7+(-4) 7+(-4)= -8-19 is the same as -8+(-19) -8+(-19)= 5-(-8) is the same as 5+8 Double negatives = + 5+8= 116-(-12)=

Multiplying Integers The product of two integers with different signs is negative X=-3(29) X=-87 Y=(-6)(-25) Y=150 If the signs are the same, the product is positive

Solve X=3(-7) Y=(-9)(18) X=-9(-5) 5X= if X=-2 (10)(-17)(5)=

Multiply you will not be able to solve for the variable Multiply you will not be able to solve for the variable. Discuss your answers as a class. -4b 8(-x) 6(-9a) 2ab(6)(-2) (-5y)(-6) 5x(-4)(-2) |-3| + |-4| =

Dividing Integers Dividing Integers with Different Signs: The answer (or quotient) of two integers with different signs is negative + - = - The quotient of two positive numbers is positive + + = + The quotient of two negative integers is positive - - = +

Perimeter The distance around a figure is called Perimeter. Add the measure of the sides = perimeter P = perimeter L= length W= width P= L+W+L+W 60in 27in

Area The measure of the surface enclosed by a geometric figure is called the area Area= Length X Width 60 in 27in

Solving Inequalities: Adding or Subtracting An inequality is a mathematical sentence with < or >. The sentences 99>82 and 124>107 are inequalities.

Solving Inequalities: Adding or Subtracting Adding the same number to each side of an inequality does not change the truth or message the inequality is saying. Example: 99 > 82 add 25 to each side Therefore: 99 + 25 > 82 + 25 Answer: 124 > 107

Solving Inequalities: Adding or Subtracting In Words: Adding or subtracting the same number from each side of an inequality does not change the truth of the inequality. In Symbols: If a > b In Symbols a + c > b + c In Symbols a - c > b – c

Solving Inequalities: Adding or Subtracting Solve a + 8 > 3 a - 8 > 3 – 8 a = - 5 Check: Try -4 a number greater than -5 -4 + 8 > 3 = 4 4 > 3 The solution is a ≥ -5, all numbers greater than -5 are possible answers. Inequalities will have multiple possible answers

Solving Inequalities: Adding or Subtracting Examples Y-7< 10 Y < 17 Because 10+7 = 17 -42 + k > 18 K > 60 +42 +42 K> 60

Solving Inequalities: Multiplying or Dividing Consider multiplying or dividing each side of the inequality 4<6 by a positive integer The Inequalities 8<12 and 2<3 are true. These and other examples suggest the following property.

Solving Inequalities: Multiplying or Dividing In Words: When you multiply or divide each side of a true inequality by a positive integer, the result remains true. In Symbols: For all integers you can think of for a, b, c: if c > 0 If A > B, then ac > bc or a/c > b/c Example: 4<6 4x2 < 6x2 8 < 12

Solving Inequalities: Multiplying or Dividing 2<4 4/2 < 6/2 2 < 3 Consider Multiplying or dividing each side of 2<4 by a negative integer 2 < 4 2 x -1 < 4 x -1 -2 < -4 FALSE The inequalities -2 < -4 is false. However, it would be true if you reverse the symbol. -2 >-4

Solving Inequalities: Multiplying and Dividing In Words: When you multiply or divide each side of an inequality by a negative integer, you must reverse the order symbol In Symbols: For all integers, a,b,c, where c<0 if a > b then ac < bc a/c < b/c Example Problems: -3y > -39 Y> 13 because -3 x 13 = -39 so the number has to be bigger than 13

Using Equations and Inequalities Believe it or not, you will use math after you graduate high school. However finding the math problems in everyday life can be tricky. We are going to learn word problems together. Ex: Pedro rides his 10-speed bike for exercise. The last time he rode his bike he had trouble changing gears. Pedro took his bike to Lewis Bike Shop to have it repaired. At the shop, Pedro was told that the total bill for labor and parts would be at least $48. The cost of the parts was $33. Question: How much could Pedro expect to pay for labor

Using Equations and Inequalities Read the problem and find the main idea. You need to find out the labor cost. Let L= labor Next, look at the information in the problem The total was at least $48 The parts cost $33 L= labor $33 + L ≥ $48 L ≥ 15

Factors Factors: The factors of a whole number divide that number with a remainder of 0. 6 is a factor or 78 because 78/6=13 a whole number You can say that 78 is divisible by six. 78/5=15.6 this is not a factor because it is not a whole number

Monomial Monomial: an integer, a variable, or a product (remember product means multiply) of integers or variables Example: 2(L+W) this is not a monomial because it has addition Example: -12abc This expression is a monomial because of integers and variables

Factor Facts A number is divisible by: 2 if the ones digit is divisible by 2 3 if the sum of its digits is divisible by 3 5 if the ones digit is 0 or 5 6 if the number is divisible by 2 or 3 10 if the ones digit is 0 Practice a few as a class

Monomial Facts Examples: y, 15, ab, 4mk, 2X+3 and m-6 are different from monomials because they have two items

Powers and Exponents An Exponent tells how many times a number (a base) is used as a factor The smaller written number is the exponent Numbers that are expressed using exponents are called powers.