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## Linear Equations and InequalitiesLinear Equations and Inequalities If the equality symbol (=) is replaced by >, <, ≥ or ≤ , the resulting
expression is called alinear inequality. Example: Example: The linear equation 3x − 8 = 23 Let’s check that for x = 1: while Also:
Step 2. What is a half? Step 3. Analyze your problem: Alex Rodriguez joined the Boys and Girls Clubs of Miami in
1982. There he “developed” as a or also, x = 100. According to our previous estimation, Is the Club going to
reach the goal? Example:
Thus we can write Example:
The recent presidential primary election for the
Democratic Party was a close one. Before one of Step 3. Expand the parenthesis. Step 4. Add .45 and .25. Step 5. Subtract .30 from both sides. Step 6. Add x * (.25) to both sides. Step 7. Divide both sides by .50. or also What percentage of Edward’s votes do Obama needs to get in
order to finish tied with Clinton?
What is a non-negative number? x≥0
On your own… On your own…HomeProblems 1 – 40 page 11 (Handout or Chapter 1, 11 ^{th} edition).Graphs and Lines. Definition: In the Cartesian Coordinate System we have two real number lines, one horizontal and one vertical. The two lines cross through their origins. The ordered pair (a,b) gives the coordinates of a point. The first coordinate (a) is called the abscissa (x-axis) and the second coordinate (b) is called the ordinate (y-axis). Example: Now construct a table:
_{1} , y_{1}) and P (x_{2} , y_{2}) then the slope is
given byExample:
On your own… Problem 62, page 27 (Handout or Chapter 1, 11 Problem 12, page 38 (Handout or Chapter 1, 11 ^{th} edition).Functions In the first chapter we learned about linear functions (lines), but not all functions are linear. Other types of functions (e.g. quadratic) are more difficult to graph. In this case we need to use point-by-point plotting. Solving a Non-linear Equation. In the first chapter we had things like: y = 9 − x
We can find a lot of the points, but we only need two
points to sketch the graph. It’s just a line!!!
On your own… Using a graph estimate how many hours McGrady needs to
practice in order to score 32 points Step 2. Setup a table:
However, points are not the only worry for McGrady. In the
year 2002-2003 he averaged 2.6
Definition: each positive real number has two square roots,
(which is called the(3)*(3) = 9principal square root) and (the negative of the principal square root). For instance, and , this because while , we
also have that(−3) * (−3) = 9 . Example: When x = 8 But since (-32)*(-32) = 1024 we also have that In general, we can write this as Is this equation a function? Since each positive real
number has two square roots to each input In general we are going to use the letter f (guess why?) to refer to a function. But we can use any letter (m, k, g, etc.). Let’s say that we have We can evaluate this function at 2 and 4: In general:
It is also common to express an equation like
as f (x) = (x + 3)
^{th} edition).
Finding a DomainLet’s say that we have the following equation: For y to be real, 10 – x must be greater than or equal to
zero.
Elementary Functions: Graphs and Transformations.
Now let’s evaluate each function at 10 and – 10.
You can shift a function vertically or horizontally. For the function f (x) =l xl g(x) = lx l+ 5 and g(x) =l xl − 5 are vertical shifts, while g(x) = lx + 5l and g(x) = lx − 5l are horizontal shifts. See Summary in page 69 (page 27, 10^{th} edition).
where b Example: Let’s verify that:
Definition: a rational function is any function that can be written in the form
Exponential Functions. Example: Homer Simpson is offering Bart and Lisa two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Bart argues that the second option, growing at a constant rate of $1/week, pays more and is obviously better. However, Lisa thinks that the first option will eventually grow much larger and selected this option. What did you think?
For Lisa, the allowance at week n is 2 Definition: The base
is the most popular of
bases. The base is an irrational for a sufficiently large x. In general = 2.71828 18284 59045 23536........
1. If the bases of the exponential expressions that are multiplied are the same, then you can combine them into one expression by adding the exponents. This makes sense when you look at: If the bases of the exponential expressions that are
divided are the same, then you When you have an exponential expression raised to a power,
you have to multiply Notice that we had to use another rule of exponents to
help us make sense of this Any number or variable raised to the zero power is always equal to 1. It is important to note that if a negative exponents
already appears in the A = amount (future value) at the end of t years.
Logarithmic form: Exponential form: x = b^{y}Example: Example: Step 2. Simplify the equation: and Or also: Step 3. Apply property 5: Step 4. Apply property 6: Step 5. Simplify the equation: Step 6. Cancel the logs |