Thinking Mathematically A review and summary of Algebra

Thinking Mathematically A review and summary of Algebra 1 By: Bryan Mc. Coy and Mike Pelant

In the next slides you will review: All of the Properties and Equations needed to succeed in the upcoming Exam

In the next slides you will review: Review all the Properties and then take a Quiz on identifying the Property Names

Addition Property (of Equality) Example: If a=b then a+c=b+c Multiplication Property (of Equality) Example: If a=b then a(c)=b(c)

Reflexive Property (of Equality) Example: If a=b then b=a Symmetric Property (of Equality) Example: If a=b then b=a (order does not matter) Transitive Property (of Equality) Example: If a=b and b=c then a=c

Associative Property of Addition Example: (1+2)+3=6 1+(2+3)=6 (Does not matter where you put the parenthesis) Associative Property of Multiplication Example:

Commutative Property of Addition Example: 5+3+2=3+5=2 Commutative Property of Multiplication Example:

Distributive Property (of Multiplication over Addition Example: 3(2+7 -5)=3(2)+3(7)+(3)(-5)

Prop of Opposites or Inverse Property of Addition Example: +8 -8=0 Prop of Reciprocals or Inverse Prop. of Multiplication Example:

Identity Property of Addition Example: 4+0=4 Any number plus 0 equals the original number Identity Property of Multiplication Example: Any number times 1 will equal itself

Multiplicative Property of Zero Example: A number times 0 equals 0 Closure Property of Addition Example: . Closure Property of Multiplication Example: Product of 2 real numbers = a real number

Product of Powers Property Example: 72 × 7 6 = 78 Power of a Product Property Example: 32 · 42 = 122 Power of a Power Property Example:

Quotient of Powers Property Example: Power of a Quotient Property Example:

Zero Power Property Example: (-3)0 = 1 Negative Power Property Example:

Zero Product Property Example: if ab = 0, then either a = 0 or b = 0 (or both).

Product of Roots Property The product of the square roots is the square root of the product. Quotient of Roots Property For any non-negative (positive or 0) real number a and any positive real number b: =√a -√b

Root of a Power Property Example: Power of a Root Property Example:

Now you will take a quiz! Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 1. a + b = b + a Answer: Commutative Property (of Addition)

Now you will take a quiz! Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 2. If a=b then a(c)=b(c) Answer: Multiplication Property (of Equality)

In the next slides you will review: Solving inequalities

Sample Problem: -5 x < 10 x > -2 Solution Set: {x: x > -2} -2 - Remember the Multiplication Property of Inequality! If you multiply or divide by a negative, you must reverse the inequality sign.

Linear Equations in 2 Variables Here’s a sample problem: can you graph this: y=x-5?

Linear Systems Can you solve this? y = 3 x – 2 y = –x – 6 Y=-x-6 3 x-2=-x-6 4 x=8 X=2 Now solve for Y Y=3(2)-2 Y=6 -2 Y=4 The answer is (2, 4)

In the next slides you will review: All of the Factoring Methods

Find GCF! Finding the GCF will make the problem simpler greatly. Ex: – 2 x-4 y=8 – GCF = (2) – = (2)(x-y=4)

The following is a list of the rest of the properties: Difference of Squares Sum/Difference of Cubes PST Reverse Foil Factor by Grouping(4 or more terms)

Rational Expressions Try this Problem:

Functions f(x)= is another way to write y= Functions are relations only when every input has a distinct output, so not all relations are functions but all functions are relations. Let’s say you had the points (2, 3) and (3, 4) and you needed to find a linear function that contained them. This is how you would do that. 3 -4 over (divided by) 2 -3 (rise over run, Y is rise, X is run) you would get -1 over -1. This equals 1, which will be the slope. To find y-intercept, substitute: 2=1(3)+b 2=3+b -1=b So your final equation is: Y=X-1. You can now graph this.

Parabolas See if you can graph this one: x 2 -6 x+5 The x-intercepts are (5, 0) and (1, 0). y-intercept: Vertex: and So the vertex is (3, -4). Now just graph it.

Simplifying Expressions With Exponents Simplify this: The "minus" on the 2 says to move the variable; the "minus" on the 6 says that the 6 is negative. Warning: These two "minus" signs mean entirely different things, and should not be confused. I have to move the variable; I should not move the 6. Your answer is:

Simplifying expressions with radicals Try this one:

Word problems You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use? Let x stand for the number of liters of 10% solution, and let y stand for the number of liters of 30% solution. (The labeling of variables is, in this case, very important, because "x" and "y" are not at all suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in the end. ) For mixture problems, it is often very helpful to do a grid:

Word Problems Continued A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3. 30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there? I'll start by picking and defining a variable, and then I'll use translation to convert this exercise into mathematical expressions. Nickels are defined in terms of quarters, and dimes are defined in terms of nickels, so I'll pick a variable to stand for the number of quarters, and then work from there: number of quarters: q number of nickels: 3 q number of dimes: (½)(3 q) = (3/2)q There is a total of 33 coins, so: q + 3 q + (3/2)q = 33 4 q + (3/2)q = 33 8 q + 3 q = 66 11 q = 66 q=6 Then there are six quarters, and I can work backwards to figure out that there are 9 dimes and 18 nickels.