An Explosion of Math!!!!

By: Matt and Nick

Quick 1st Power Equation

Example: 4x=12

Answer: x=3

Special Cases of These Equations

• A. x3-7x2=-6x

-6x=-6x= (All real #’s)• B. 5x/3 + 7/2 = 4

6*5x/3 + 6*7/2 = 6*410x+21 = 2410x = 24-2110x = 3x = 3/10

• C. 4/x=12 x=3

Addition Property (of Equality)

Multiplication Property (of Equality)

Example: If x = y, then x + z = y + z.

If a+2=7, then a+2+-2=7+-2

Example: If a = b, then a * c = b * c

Reflexive Property (of Equality)

Symmetric Property (of Equality)

Transitive Property (of Equality)

Example: 3m=3m

Example:If m=n, then n=m

Example: If m=n and n=p, then m=p

Associative Property of Addition

Associative Property of Multiplication

Example:(7+1/4)+3/4=7+(1/4+3/4)

Example: a(bc) = (ab)c

Commutative Property of Addition

Commutative Property of Multiplication

Example: 1/4+7+3/4=1/4+3/4+7 

Example: ab = ba

Distributive Property (of Multiplication over Addition)

Example:If -3(x-2)=1, then -3x+6=1 

Prop of Opposites or Inverse Property of Addition

Prop of Reciprocals or Inverse Prop. of Multiplication

Example: a+(-a)=0

Example:-3/x*-x/3=1

Identity Property of Addition

Identity Property of Multiplication

Example: 0 + a = a = a + 0

Example: 1 * a = a = a * 1

Multiplicative Property of Zero

Closure Property of Addition

Closure Property of Multiplication

Example: a × 0 = 0

Example: If x and y are real numbers, then x+y is a real

number.

Example: If x and y are real numbers, then x*y is a real

number.

Product of Powers Property

Power of a Product Property

Power of a Power Property

Example: ab × ac = a(b + c)

Example: (ab)m = am · bm

Example: (ab)c = abc

Quotient of Powers Property

Power of a Quotient Property

Example:

Example:

Zero Power Property

Negative Power Property

Example: 170 = 1

Example: x-3=1/x3

Zero Product Property

Example: If ab = 0, then either a = 0 or b = 0 (or

both).

Product of Roots Property Example:

Quotient of Roots PropertyExample:

5 5 25

9 39

x x x

Root of a Power Property

Power of a Root Property

Example:

Example:

3 3x x

2

x x

Quiz Time!!!

***You will see an example problem and you will click to see the answer! There are 10 Problems so it should

only take a few minutes to complete. Have Fun!

x9*x3=x12

Product of Powers Property

(xy)3= x3y3

Power of Product Property

x3=x3

Reflexive Property of Equality

x3*0=0

Multiplicative Property of Zero

If x-3=9, then x-3+3=9+3

Addition Property of Equality

If x and y are real numbers, then x+y is a real number.

Closure Property of Addition

x3*1=x3

Identity Property of Multiplication

(x9)3=x27

Power of a Power Property

9(x-y)=9x-9y

Distributive Property

y3x=xy3

Commutative Property of Multiplication

First Power Inequalities

***In the following slides you will see how to solve first power

inequalities.

One Inequality Sign

Answer: X<3

X+3<6

***To answer this, you would subtract 3 from both sides and end up isolating the variable on the left side and 3 on the other. The inequality sign would stay the same because you are not multiplying/dividing by a negative number.

Conjunction

Answer: -2<X<3

-2<x and x<3

***To solve a conjunction of two open sentences in a given variable, you find the values of the variable for which both sentences are true.

Disjunction

Answer: y<-3 or y>7

y-2<-5 or y-2>5

***To solve a disjunction of two open sentences, you find the values of the variable for which at least one of the sentences is true.

All Real #’s

Answer: {All real Numbers}

n+5 n+5

***As you can see, the inequalities cancel out to leave a technically true statement leaving the answer to be “All real numbers”

No Solution

Answer: No Solution

***Two inequalities have no solution when both of them must be true and they result in mutually exclusive conditions. Thus, there is no number that is both greater than 5 and less than 3, therefore there is no solution.

x + 5 > 10 and x -2 < 1

How To Do Linear Equations

• Slopes of All Lines:• Rising line-positive slope• Falling line-negative slope• Vertical line- undefined • Horizontal line- 0

• Equations of All Lines• Horizontal- y=c • Vertical- x=c • Diagonal- y=mx+b and Ax+By=C

Linear Equations Cont.

• Standard/general form: Ax+By=C

• Point-slope form: y-y1=m(x-x1)

• Slope intercept form: y=mx+b

• How to Graph: Video from Math TV

• Click here to Graph y=3x-1

Linear Equations Cont.

• How to Find Intercepts

1.Put the equation into Slope-Intercept form

2.Y=mx+b

3.The “b” in the equation is your Y-intercept

Linear Systems

Substitution Method

1. Solve the first equation for y

2. Substitute this expression for y in the other equation, and solve for x.

3. Substitute the value of x in the equation in Step 1, and solve for y.

***P.417 in your book has great examples!

Elimination Method

1. Add similar terms of the two equations

2. Solve the resulting equation

3. Substitute what you got for x and plug it into either of the equations and solve for y

***P.426 in your book has great examples!

Systems of Equations

• Independent- two distinct non-parallel lines that cross at exactly one point (solution is always some x,y-point)

• Dependant- two lines that intersect at every point (solution is the whole line)

• Inconsistent- shows two distinct lines that are parallel (never intersect), has no solution

• ***Graphs of these terms are on following slide!

Graphs from www.purplemath.com

Independent system:one solution and

one intersection point

Inconsistent system:no solution and

no intersection point

Dependent system:the solution is the

whole line

                                               

                                              

 

                                              

 

Factoring

• Grouping (2x2 and 3x1)- You use this when you have 4 or more terms

• GCF- You use this when you have any number of terms

• Difference of Squares- Use this with Binomials• Sum and Difference of Cubes- Use with

Binomials• PST- Trinomials• Reverse FOIL-Trinomials

Rational Expressions

*Factor first!

Answer

Factor and Cancel

*Common factor in both the numerator and the denominator and so we can cancel the x-4 from both

Rational ExpressionsAddition and Subtraction of Rational

Expressions

*Common denominator is: 6x5

*Multiply each term by an appropriate quantity to get this in the denominator and then do the addition and subtraction

Answer

Rational ExpressionsMultiplication of Rational

Expressions

Answer

*The first thing that we should always do in the multiplication is to factor everything in sight as much as possible

*Cancel as much as we can and then do the multiplication to get the answer

Rational ExpressionsDivision of rational expressions

Answer

*Divide first!

*Once we’ve done the division we have a multiplication problem and we factor as much as possible, cancel everything that can be canceled and finally do the multiplication.

Quadratic Equations in one Variable

• Quadratic –second power

• Use the discriminant to predict how many x-intercepts each parabola will have.

Solve by FactoringSo the first thing to do is factor:

x2 + 5x + 6 = (x + 2)(x + 3)

Set this equal to zero:

(x + 2)(x + 3) = 0

Solve each factor:

x + 2 = 0  or  x + 3 = 0

x = –2  or  x = – 3

Answer: x2 + 5x + 6 = 0 is x = –3, –2

Quadratic Equations in one Variable Cont.

Taking the square root of each side:

m2=49

m= √49

m= 7

Method of Completing the Square

For: x2+bx+?

1. Find half the coefficient of x: b/2

2. Square the result of step 1: (b/2)2

3. Add the result of step 2 to x2+bx: x2+bx+(b/2)2

4. You have completed the square: x2+bx+(b/2)2=(x+b/2)2

Quadratic Equations in one Variable Cont.

Complete the square:

X2+14x=?

X2+14x+49= (x+7)2

(14/2)2=49

Quadratic Equations in one Variable Cont.

Quadratic Formula:

x=-b+√b2-4ac/2a

Line of Best Fit or Regression Line

• We use this to find an equation for a scatter plot.

• Your calculator will help you find the best fit line

• The calculator will find an exact regression line

Line of Best Fit or Regression Line Cont.

Write an equation of a line that has a slope of -4 and x-intercept of 3.

1. Substitute -4 for m in y=mx+b2. To find b, substitute 3 for x and 0 for y in y=-

4x+by=-4x+b0=-4(3)+b0=-12+b

12=bFinal answer: y=-4x+12

Functions

• A. f(x) means "y“ and not all functions are relations• B. A function can only use each x-value once

– Domain- set of all x-coordinates (independent)– Range- set of all y-coordinates (dependant)

• Find the range given f(x)=5x-3 and Domain={-2,0,7}.    f(-2)=5(-2)-3=-13    f(0)=5(0)-3=-3    f(7)=5(7)-3=32Range={-13,-3,32}

• C. We will show hot to do a Parabola on the next slide

How to Graph a Parabola

1. The easiest way to graph a parabola is to start by finding the x-coordinate of the vertex, or the turning point of the function. Given a parabola with a general equation of y=ax²+bx+c, the x-coordinate of the vertex can be found by using x=-b/2a, which is the equation of the axis of symmetry for the parabola. The axis of symmetry runs through the vertex, and therefore shares a common point.

2. Substitute the x value of the vertex back into the function to find the y value of the vertex.

3. After you've found your turning point, you can select two x-values to the right of the turning point and two values to the left of the turning point.

4. Plot all points

Simplifying Expressions with Exponents

1. x6 × x5 = (x6)(x5)              = (xxxxxx)(xxxxx)    (6 times, and then 5 times)             = xxxxxxxxxxx         (11 times)              = x11  

2. Simplify (–46x2y3z)0 This is simple enough: anything to the zero power is just 1.

(–46x2y3z) =1

3.

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.

*** The answer is -6x2

2

6

x

Simplifying Expressions with Radicals

1. Simplify

2. Simplify

3. Simplify

54

54 9 6 3 6

3 3203 3 3320 64 5 4 5

4 2a b4 2 2a b a b

Word Problems

1. Problem: The sum of twice a number plus 13 is 75. Find the number.

Hint: The word is means equals. The word and means plus. Therefore, you can rewrite the problem like the following:

The sum of twice a number and 13 equals 75.

Solution: 2N + 13 = 75

N=31

Word Problems Cont.

2. At the same moment, two trains leave Chicago and New York. They move towards each other with constant speeds. The train from Chicago is moving at speed of 40 miles per hour, and the train from New York is moving at speed of 60 miles per hour. The distance between Chicago and New York is 1000 miles. How long after their departure will they meet?

x/40 = (1000-x)/60, we can simplify it as x/40 + x/60 = 1000/60, or

x = 40 * 1000/(40 + 60) = 400

*The time that it takes the train from Chicago to travel 400 miles, is x divided by the speed of the Chicago train, which is t = x / 40 = 10.

***So, the answer is: 10 hours.

Word Problems Cont.

3. The total receipts for a hockey game are $1400 for 788 tickets sold. Adults paid $2.50 for admission and students paid $1.25. How many of each kind of tickets were sold?

• Alright, let's denote the quantity of ADULT tickets sold as x. Since the total number of tickets is The quantity of student tickets was 788, the number of student tickets is 788-x. What we need to do now is write the total $$ figure for the revenue. Since every adult ticket fetched 2.50, adults collectively have paid x*2.50. The students paid 1.25, and since we had 788-x of them, they paid the sum of (788-x)*1.25. So, we have total revenue = x*2.50 + (788-x)*1.25

• What we know is that the total revenue for the basketball game was 1400. This gives us the equation

• 1400 = x*2.50 + (788-x)*1.25

• Rewriting, we get

• x*(2.50-1.25) = 1400 - 788*1.25 or x*1.25 = 415 or x = 415/1.25 or x = 332

• That's the number of adult tickets. The number of student tickets is, therefore, 788-x, or 456. That's it!

***The answer is 332 adult tickets and 456 student tickets!

Word Problems Cont.

• Alright. So, we have 4(x-4) = 3x-4 or 4x - 4*4 = 3x-4 or, moving everything x-related to the left and numbers to the right, 4x - 3x = 4*4 - 4 x(4 - 3) = 4*4 - 4 or, dividing by 4 - 3: x = (4*4 - 4) / ( 4 - 3 ) or, calculating x, x = 12

***Bob's age is 12. His father is 36 years old.

The End!

Hope you enjoyed the show and have a great summer vacation!