FNMT 118.Linear Eqns

Translate the following into algebraic expressions...

• y divided by 4
• x less than 5
• two less than the product of 10 and Goran's savings
• three more than the product of a number and 5 is equal to 7

FNMT 118

Linear Equations & Inequalities

Linear Equations in One Variable

Distinguish between expressions and equations

Identify Linear Eqns & Whether a Number is a Solution

Solve Linear Equations with Addition & Multiplication Properties of Equality

Exponents

Linear Inequalities

• Linear equation is a first-degree equation, b/c the greatest power on the variable is 1

• If the variable in an equation can be replaced by a real number that makes the statement true, then that number is a solution of the equation

• An equation is solved by finding its solution set, the set of all solutions

Speed Round: Arithmetic Review

Linear Inequalities... what are they?

Product Rule

Solve Linear Equations Using Distributive Property

Solve linear equations with fractions & decimals

Conditional Equations, Contradictions, & Identities.

Solutions to & Solving Linear Inequalities

a) Write an inequality for the following statement

b) Graph the inequality

c) Write the solution in set-biilder notation and interval notation

• x is less than or equal to 8
• 4 is greater than or equal to 4

a) Graph the compound inequality on the number line.

b) Write the solution in set-biilder notation and interval notation

How would you write "x cubed"?

What about "x cubed times x to the fourth"?

Quotient Rule

• Determining the solution to a linear equation and inequality are the same.
• Plug in the potential solution and determine whether the result is true or false.

True = solution

False = not a solution

Let's try...

-5 (x + 1) + 3x + 2 = 6x + 4

2 (x + 3) = -4 (x + 1)

4 [6 - (1 + 2x)] + 10x = 2 (10 - 3x) + 8x

To solve Linear Equations we use the addition and multiplication properties of equality... what if they were inequalities instead?....

Why can we solve x + 5 = 8 by subtracting 5 from both sides?

How does this translate to x + 5 < 8?

Does it work with multiplication and division?

Simplify Exponential Expressions

What is "two to the fifth divided by two squared"? Why?

Apply this rule to the following

Compound Inequalities

Exponents, Roots, & Order of Operations

Compound inequalities incorporate the concepts of 'and' & 'or'

In an 'and' statement both conditions must be true for something to be included in the solution

In an 'or' statement, one or both conditions can be true.

Try these...

Operations on Real Numbers

Square Roots

Distance Between Two Points

Literal Equations & Translating

Exponents

How are exponents and square roots related?

Subtract Real Numbers

Add Real Numbers

What are exponents? What do they mean?

Multiplication is repeated edition. How is this related to exponents?

Solve for Specified Variable

Write using exponents and evaluate.

Find each square root that is a real number.

What is the distance between the points 8 and 2? What about 2 and 8?

Find the distance between points -12 & -1.

Let's do some practice with exponents and different signs.

Find the sum.

-3 + 17

-6 + (-15)

-1.1 + 0.7

-3.8 + 4.6

Zero & Negative Exponents

When solving for a specified variable, the key is to treat that variable as if it were the only one. Treat all other variables like numbers (constants).

Solve the formula d = rt for r

Solve the formula for L.

P = 2L + 2W

Substitution & Evaluation

Order of Operations

Power Rules

Solve the equation for x.

Reciprocals & Divide Real Numbers

What is 3^0 or 3^-2?

Let's look at the pattern of exponents to figure it out

Multiply Real Numbers

Translating into Linear Equations

a is 1/a

A total of 300 tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was three times the number of adult tickets sold. How many adult tickets were sold?

Devaughn's age is three times Sydney's age. The sum of their ages is 72. What is Sydney's age?

For his long distance phone service, Greg pays a $5 monthly fee plus 9 cents per minute. Last month, Greg's long distance bill was $15.44. For how many minutes was Greg billed?

The definition of division depends on the idea of a multiplicative inverse, or reciprocal.

A number and its additive inverse have opposite signs. However, a number and its reciprocal always have the same sign.

How could we explain this for every case?

Power to a Power

Product to a Power

Let's look at (4^3)^5...

Reciprocals have a product of 1.

How about (3*5)^4?

Division by 0 is undefined, whereas dividing 0 by a nonzero number gives the quotient 0.

Quotient to a Power

Aaaaand... (2/3)^4?

Factoring

Greatest Common Factors & Factoring by Grouping

What does "factoring" mean?

Recall, if you will, reducing fractions... like 3/15 or 7/21

In order to reduce we need something called the greatest common factor (GCF)

GCF - is the largest number that is a factor of all the terms

Roots & Radicals

Factor by grouping

Usually done when a polynomial has more than 3 terms (sometimes we have to manipulate it to have more than 3 terms... hint for future use)

6p – 6q + rp – rq xy – 2y – 4x + 8 kn – m – k + mn

Factor out the GCF

How this applies to polynomials?

Polynomials

Factoring Trinomials

Polynomials can have a GCF as well...

7k + 28 = ?

How can you check?

32m + 24 = ?

Factor trinomials when the coefficient of the quadratic term is 1

Think about how we got to trinomials and what factoring means. What should our final answer look like?

Factor trinomials when the coefficient of the quadratic term is not 1.

Find roots of numbers

This one is tricky! How can I think about it to make it easier on myself?

Maybe think of one of the variables as a constant?

Let's think about the signs... how will that effect our process, answer, etc.?

Multiplying Polynomials

Special Factoring

Simplifying Radicals

Multiply two binomials

Use an alternative method for factoring trinomials

What is FOIL really?!?!?

And when can you use it?

Steps

1. GCF

2. AC

3. B = Factors

4. Factor by Grouping

Factor a difference of squares.

Quotient rule for radicals

Product Rule for radicals

Difference of squares looks awfully familiar... but from where?

Find principal roots

What is the square root of 9 times the square root of 25?

Is that the same as the square root of (9 times 25)?

Simplifying radical expressions

And for fun...

What is sqrt(16/25)?

What about sqrt(16)/sqrt(25)?

Multiply monomials

Binomial Squared

Adding & Subtracting Polynomials

Where have we seen this already?

Try the following...

What very common mistake are we going to avoid?

What laws or rules do we know that can help us?

Find all square roots of...

• 25
• 4
• 49
• -144

If there is more than one answer,

separate them with commas.

If there are none,

click on "None".

Factor a perfect square trinomial.

Now if I recognize some relationships in this polynomial it begins to look familiar... but from where?

Multiply coefficients separately

Multiply appropriate variables using the exponent laws

Polynomials

Making it even more fun...

• Polynomial = term +/- term
• Term = a number (constant), a variable, or the product or quotient of a number and one or more variables raised to powers

Adding polynomials...

Multiply any two polynomials

Product of conjugates

A General Approach to Factoring

It's combining like terms... shhh don't give away the secret.

1. Factor out any common factor.

3. Factor trinomials.

Radicals in Solving Quadratic Equations

Not all quadratics can be solved by factoring... some need either the square root or the quadratic formula.

2. Factor binomials.

Solving Equations

by Factoring

4. Factor polynomials of more than three terms.

Dividing Polynomials

Polynomial divided by a polynomial of two or more terms

Square root property

Learn and use the zero-factor property

Quadratic Formula

When solving a quadratic using the square root property you have to remember the principle root and negative roots are both possible solutions.

Not all quadratics can be solved by factoring.... you may need the quadratic formula

Polynomial divided by a monomial

Solve (8x + 3)(2x + 1) = 0..... think about xy = 0... what do we know?

Check the solutions.

Solving Quadratic Equations

Remember... 2x = 10 + 6

Fill in the missing info...

Zero-Factor Property

If two numbers have a product of 0, then at least one of the numbers must be 0.

That is, if ab = 0, then either a = 0 or b = 0.

You can tell what kind of solution(s) you will get by using the discriminant...

Compute the value of the discriminant and give the number of real solutions of the quadratic equation.

5x^2-6x+3=0