+1 862 207 3288 MODULE 1
Fill in the following blanks and submit using the assignment tool in Module 1. You can fill in the blanks and submit as a word document. The second option is to print out the form, write the answers on the printout, scan your answers and submit your work as a PDF file.
Real numbers
Natural numbers: the counting numbers, N = {1,2,3,4,5,…}
Whole numbers: 0 and the counting numbers, W = {0,1,2,3,4,…}
Integers: whole numbers and their opposites, Z = {…-2,-1,0,1,2,…}
Rational numbers: can be expressed in the form p/q, such that p and q are _______________.
The symbol ________ is often used to represent the set of rational numbers.
All rational numbers can also be expressed as either ______________ decimals or _________________ decimals.
Which of the following are irrational numbers? {3.24,-7,√5,√36,π,7/(-3)} ____________________
Set-Builder Notation and Interval Notation
{x|x<-5} is __________________ notation and is read: “_________________________________________________________”.
Types of intervals
Open intervals: (a,b) does not include the endpoints a and b.
Closed intervals: [a,b] does include the endpoints a and b.
Half-open: ____________ does not include a but does include b.
Infinite interval: (-∞,4) is equivalent to {x|______________}.
Definition of Absolute Value: The absolute value of a real number a is defined by |a| = ______ if a > 0 and |a| = ______ if a < 0.
Order of Operations
Start within the innermost ___________________ symbols. This includes parentheses ( ), ______________ [ ], braces { }, and _________________________ bars | |.
Simplify _________________ expressions.
Perform __________________ and ________________ from left to right.
Perform __________________ and ________________ from left to right.
Laws of Exponents
b^(m⁄n)=√(n&b^m )=〖(√(n&b))〗^m
b^(-s)= 1/b^s
b^0= ________
〖16〗^(-3⁄4)= __________
Polynomial Terminology
An algebraic expression with one term in which the exponents of all variable factors are nonnegative integers is called a _________________.
The _____________________ of a monomial is the numeric factor.
The degree of a monomial is the sum of the variable ________________.
A polynomial with two terms is called a _____________________.
A polynomial with three terms is called a _____________________.
Special Factoring Formulas
Difference of Two Squares: a2 – b2 = (a + b)(_________)
Sum of Two Cubes: a3 + b3 = (________)(a2 – ab + b2)
Factor: 27×3 – 8 = (________)(9×2 + _____ + ______ )
Rational Expressions
A rational expression is the quotient of two __________________ expressions.
Subtract: (x-2)/(x+4)-(x+1)/(x-4)
Multiply the numerator and denominator of the first fraction by _______ and multiply the numerator and denominator of the second fraction by _________.
This gives: (x^2-6x+8)/((x+4)(x-4))-(x^2+5x+1)/((x-4)(x+4))
Subtracting the numerators gives (-11x+7)/((x+4)(x-4))
Solve Linear Equations Involving Fractions or Decimals
Solve 6/(x^2-x)-2/x=3/(x-1)
Factor the denominators. x2 – x = _____________
The Least Common Denominator is _______________.
Multiply both sides of the equation by the LCD
x(x-1) 6/x(x-1) -x(x-1) 2/x=x(x-1) 3/(x-1)
Simplifying each term by cancelling gives:
6-(x-1)2=3x
Solving this equation for x gives x = ______.
Solve .12x + .3(x-4) = .01(3 – x)
Multiply both sides of the equation by ______ to remove the decimals
This gives 100(.12x) + 100(.3)(x-4) = 100(.01)(3 – x)
Simplifying each term gives: 12x + 30x – 120 = 3 – x.
Solving this equation for x gives x = _______.
Applications
Strategy for Solving Applications
_________ the problem several times. Create tables or diagrams to assist your understanding and organize your thoughts.
Assign a ______________ to the unknown value you are being asked to find. Write the other quantities in terms of this variable. Write an ____________________ .
Solve the equation.
Interpret and check your answer.
Mixture Problem
1. (%)(liters of mixture) + (%)(liters of mixture) = (%)(liters of mixture)
Starting amount + amount added = final amount
Use this model to write an equation for the following problem. Bill needs 10% hydrochloric acid for a chemistry experiment. How much 5% acid should he mix with 60 milliliters of 20% acid to obtain a 10% solution?
(_____)(60) + (______)(x) = (.10)(____ + x )
Complex Numbers
Terminology
The imaginary unit is ______, which is defined as the square root of _______. Thus _____ = -1.
In the complex number a + bi, a is called the _________ part and ______ is called the imaginary part.
Operations with Complex Numbers
Add: (3 + 2i) + (2 – 5i) = ______________.
Subtract: (3 + 2i) – (2 – 5i) = ______________.
Multiply: (3 + 2i)(2 – 5i) = _____________.
Divide: (3 + 2i)/(2 – 5i) = _____________.
(a + bi)(_________) = a^2+b^2.
Simplifying Radicals
If a>0,”then ” √(-a)=____√a.
Simplify the following:
√(-20) = _________.
√(-2) √(-8) = _________.
Quadratic Equations
Zero Product Property
If ab = 0, then _____ = 0 or _____ = 0 or both equal 0.
If (x-2)(x+3)=0 “then” x = ______ or x = _______.
Square Root Property
The solution to the equation x^2-k=0 “or ” x^2=k “is ” x=±√k.
Solve x2 – 25 = 0. The solutions are _____ and _____.
Solve (x-2)^2=9.
Taking the square roots of both sides gives: x – 2 = + ______
Add 2 to both sides of the equation. [ x – 2 + 2 = + 3 + 2 ]
The solutions are _____ and _____.
Quadratic Formula
The solutions of the quadratic equation ax^2+bx+c=0, where a ≠ 0, are given by x=(-b±√(b^2-4ac))/2a.
The expression b^2-4ac is called the _______________________.
If the discriminant is positive, there are ______ real solutions.
If the discriminant is negative, there are ______ nonreal solutions.
If the discriminant is zero, there is _______ real solution.
The solutions to the quadratic equation x^2+8x+13=0, are _____ ±√3.
Use Quadratic Equations to Solve Distance, Rate and Time Applications
A boat traveled downstream a distance of 45 mi and then came right back. If the speed of the current is 5 mph and if the total trip took 3 hours and 45 minutes, find the average speed of the boat relative to the water.
The unknown value we are asked to find is the _______________ of the boat. Let s represent this quantity.
We can represent the rate downstream as the speed of the boat plus the speed of the current, or s + 5. We can represent the rate upstream as the speed of the boat minus the speed of the current, or __________.
Since rate x time = distance, we know distance/rate=time”.” This produces the following table:
Distance Rate Time
Downstream 45mi s + 5 45/(s+5)
Upstream 45mi s – 5 45/(s-5)
Since the total time of the trip (downstream plus upstream) is 3 hours and 45 minutes (or 3 3/4 hours), the equation is:
45/(s+5)+45/(s-5)= 15/4
The LCD of this equation is ____________________.
Multiplying both sides by the LCD and simplifying each term produces: 180(s-5)+180(s+5)=15(s+5)(s-5)
Putting this equation into standard form gives: ____________________ = 0
The average speed of the boat is _______ mph.
Solving Equations that are Quadratic in Form
Procedure
Find a variable expression in the equation that is “disguised” as being raised to the first power in one place and raised to the second power in another place. In the equation (1/(x-2))^2+2(1/(x-2))-15=0, we find the variable expression 1/(x-2) raised to the ______ power in the first term and raised to the _______ power in the second term.
Assign a different variable to the variable expression. In the above example, let u=1/(x-2).
Substitute this different variable into the equation for the variable expression. In our example this produces the equation u2 +2u -15 = 0.
Solve this quadratic equation. In this example u = ____ or _____.
Set the variable expression equal to these solutions. 1/(x-2)=-5 or 1/(x-2)=3
Solving these two equations for x gives us x = ____ or _____.
Solving Radical Equations
Procedure with one radical
Isolate the term with the radical
Raise both sides of the equation to the power that will eliminate the radical.
If it is √, then square both sides
If it is ∛, then cube both sides
If it is √(n&), then raise both sides to the nth power.
Solve the resulting equation
Test the solutions in the original equation
If there are two radicals, isolate and eliminate one of the radicals. Then isolate and eliminate the other radical.
Example
Solve √(x-5)+7=x
Isolate the radical: √(x-5) = ____________
Square both sides: __________ = x2 – 14x +49
Put the quadratic equation into standard form: 0 = x2 – 15x + 54
Solve the quadratic equation for x: x = _____ or _____
Test both solutions in the original equation: Which solutions are correct? x = ________
Linear Inequalities
To solve a three part inequality, simplify it until the variable is “sandwiched” in the ________________.
Solve -3<1-2w<4
a. Add -1 to each of the three parts: -4 < _______ < 3
b. Divide each of the three parts by -2: 2 ____ w ____ -3/2
c. Write the solution in interval notation: (-3/2,2)
B. Compound Inequalities are two inequalities that are joined together by the words “or” or “_______”.
1. A number is a solution to a compound inequality involving or, if it is a solution to _______________ inequality.
2. A number is a solution to a compound inequality involving and, if it is a solution to _______________ inequalities.
3. Solve 2x – 3 < 5 and 5x + 1 > 6:
a. Solving 2x – 3 < 5, gives x < ______
b. Solving 5x + 1 > 6, gives x > _______
c. Thus x < 4 and x > 1. Which numbers would these be? Answer in interval notation: _____________
Absolute Value Equations and Inequalities
A. Properties. Given that c is a positive number:
1. |u| = c is equivalent to: u = c or u = _____
2. |u| < c is equivalent to: ______ < u < _______
3. |u| > c is equivalent to: u > ______ or u < ______
B. Solve |6x + 7| – 8 = 3
1. Isolate the absolute value: |6x + 7| = ______
Using Property 1, 6x + 7 = _____ or 6x + 7 = _____
Solving both inequalities, x = _____ or ______
C. Solve |5x – 1| + 7 < 9
Isolate the absolute value: |5x – 1| < _____
Using Property 2, ______ < 5x – 1 < ______
Solving the three part inequality gives, in interval notation: _______________
D. Solve |x + 4| > 5
1. Since the absolute value is already isolated on the left side of the inequality, go directly to Property 3: x + 4 > _____ or x + 4 < _____
2. Solving the compound inequality gives, in interval notation
_________ ∪ _________
Solving Polynomial Inequalities
A. Steps for Solving
Move all terms to one side of the inequality leaving _______ on the other side.
____________ the nonzero side of the inequality.
Find all boundary points by setting the factored polynomial equal to _______.
Plot the boundary points on a number line. If the inequality is < or > , then use a ___________ circle ●. If the inequality is < or >, then use an _________ circle ○.
Now that the number line is divided into intervals, pick a _______ value from each interval.
_____________ the test value into the polynomial, and determine whether the expression is ______________ or ______________ on the interval.
Determine the ______________ that satisfy the inequality.
B. Solve 3×2 + x < 3x + 1
1. 3×2 – _____ – 1 < 0
2. (3x + 1)(x – 1) < 0
3. The boundary values are ______ and _______.
4. Plot the boundary points on the number line
5. Pick a test value from each of the three intervals. The intervals on the number line are: (-∞,-1/3), (-1/3, 1) and (1,∞).
a. Let’s choose x = -1 in the first interval, x = 0 in the second interval, and x = 2 in the third interval.
b. Substitute these values for x in step 1 or step 2 above:
i. if x = -1, then we get 4 < 0
ii. if x = 0, then we get -1 < 0
iii. if x = 2, then we get 7 < 0
Since the second test value, 0, was the only one to give us a true statement, only the second interval is in our solution, (-1/3, 1).
Solving Rational Inequalities
Steps for Solving
Move all terms to one side of the inequality, leaving ________ on the other side.
Combine the terms to get one fraction.
Factor the nonzero side of the inequality.
Find all boundary points by setting the factored polynomial in the numerator and the denominator equal to ________.
Plot the boundary points on a number line. If the inequality is < or > , then use a ___________ circle ● for the boundary points of the numerator (but never let the denominator equal zero). If the inequality is < or >, then use an _________ circle ○.
Now that the number line is divided into intervals, pick a _______ value from each interval.
_____________ the test value into the polynomial, and determine whether the expression is ______________ or ______________ on the interval.
Determine the ______________ that satisfy the inequality.
Solve 4/(x+1)≤2
Move the 2 to the left side: 4/(x+1)-2≤0
Combine the terms on the left side by changing the -2 to (-2(x+1))/(x+1) : (2-2x)/(x+1)≤0
Factor the left side: (2(1-x))/(x+1)≤0
The boundary points are _____ and ______
Plot the boundary points on a number line. Use a closed circle for the boundary point of the numerator, but not the denominator.
Pick a test value from each of the three intervals. The intervals on the number line are: (-∞,-1), (-1, 1) and (1,∞).
a. Let’s choose x = -2 in the first interval, x = 0 in the second interval, and x = 2 in the third interval.
b. Substitute these values for x in step 1, 2 or 3 above:
i. if x = -2, then we get _____ < 0
ii. if x = 0, then we get 2 < 0
iii. if x = 2, then we get ______ < 0
Since the first and third test values gave us true statements, the first and third intervals contain our solutions. In interval notation we write: ( ________ ) ∪ [ ________ ).
#4
MODULE 2
Fill in the following blanks and submit using the assignment tool in Module 2. You can fill in the blanks and submit as a word document. The second option is to print out the form, write the answers on the printout, scan your answers and submit your work as a PDF file.
Find the Midpoint of a Line Segment
To find the midpoint of a line segment, we simply ______________ the x- and y-coordinates , respectfully.
The midpoint of the line segment with endpoints (x_1,y_1 ) “and ” (x_2,y_2 ) is ((x_1+x_2)/2,(y_1+y_2)/2).
The midpoint M of the segment with endpoints (-3,-3) and (1,4) is ______________.
Finding the Distance between two points
The distance formula is based on the _____________________ Theorem.
The distance between A(x_1 〖,y〗_1 )” and ” B(x_2,y_2 )” is given by the formula:”
” ” d(A,B)=√((x_2-x_1 )^2+(y_2-y_1 )^2 )
Find the distance between P(-3,4) and R(2,-1)
Letting x1 = -3, x2 = 2, y1 = 4, and y2 = ______, we get: d(P,R)=√((2-[-3])^2+(-1-4)^2 )
Simplifying gives: √(5^2+〖(-5)〗^2 )=√50= _____√2.
III. Standard Form of the Equation of a Circle
A. A circle is the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the ____________ of the circle and the fixed distance is called the ______________ of the circle.
B. The standard form of an equation of a circle with center __________ and radius ______ is (x-h)^2+(y-k)^2=r^2
C. Find the standard form of the equation of a circle with center (2, -5) and radius 4.
1. In the standard form given in B, h = 2, k = -5 and r = 4 giving:
(x – _____ )2 + (____ + 5)2 = _______
D. Given the equation of a circle is (x – 4)2 + (y + 3)2 = 36:
1. The center of the circle is __________
2. The radius of the circle is _________.
3. To find the x-intercepts, set y = 0 and solve for x
a. (x – 4)2 + (0 + 3)2 = 36
b. x2 – 8x + 16 + _____ = 36
c. x2 – 8x – 11 = 0
d. x = (8±√(64-4(-11)))/2 = ________ or ___________ [round to nearest hundredth]
E. The _______________ form of the equation of a circle is: Ax2 + By2 +Cx +Dy + E = 0
F. Write the equation of the circle x2 + y2 – 2x + 6y + 1 = 0 in standard form and find the center and radius
1. Group the x-terms and y-terms on the left side of the equation:
x2 – 2x + y2 + 6y = -1
2. Complete the square with both the x-terms and the y-terms:
x2 – 2x + ____ + y2 + 6y + _____ = -1 + 1 + 9
3. Factoring the left side and simplifying the right side gives (x – 1)2 + (y + 3)2 = 9
4. The center of the circle is _________ and the radius is _________.
IV. Slope of a Line
A. A line going up from left to right has _____________ slope. A line going down from left to right has __________________ slope. Horizontal lines have _________ slope, and vertical lines have ________________ slope.
B. The slope can be computed by comparing the vertical change, or the _________, to the horizontal change, or the __________. The slope of a line through the points (x1, y1) and (x2, y2) is given by m=rise/run=(change in y)/(change in x)=(y_2-y_1)/(x_2-x_1 )
C. Find the slope of the line containing the points (3, -5) and (-2, 7). m = __________
V. Forms of the equation of a line
A. The point-slope form of the equation of a line is __________________________.
B. The slope-intercept form of the equation of a line is _______________________.
C. The standard form of the equation of a line is _________________________.
VI. Writing Equations of Lines
A. Write the equation of the line containing the point (-2, 1) and having slope 3:
1. y – 1 = ____ (x + _____)
2. Changing to slope-intercept form gives y = 3x + _____ .
B. Given the graph of a line, write the equation of the line in slope-intercept form:
y = _________________
C. Write the equation of the line containing (-1,2) and (3,5) in slope-intercept form.
1. The slope of the line (use the formula in IV. B above) is: ___________
2. Using the point (-1, 2) and the slope, write the equation in point-slope form:
y – ______ = (3/4)(x + ______)
3. Simplifying gives y = (3/4)x + ________.
D. The equation of a _______________________ line has the form y = b.
1. The equation of the horizontal line through the point (-4, 7) is ___________.
E. The equation of a _______________________ line had the form x = a.
1. The equation of the vertical line through the point (-3, 5) is ____________.
VII. To find the x-intercept of a line set ______ equal to zero and solve for _______. To find the y-intercept of a line set ______ equal to zero and solve for ______.
A. Find the x- and y-intercepts of the line given by 3x – 5y = 10.
1. To find the x-intercept, set y = 0: 3x – 5(0) = 10
2. Solve this equation for x; x = ______
3. To find the y-intercept, set x = 0: 3(0) – 5y = 10
4. Solve this equation for y: y = ______
VIII. _________________ lines are lines that never intersect, _____________________ lines intersect at a right angle.
A. Two distinct nonvertical lines in the Cartesian plane are parallel if and only if they have the __________ slope.
B. Two nonvertical lines in the Cartesian plane are perpendicular if and only if the product of their slopes is ________.
C. If the slope of line l1 is a/b:
1. l2 is parallel to l1 if the slope of l2 is ________.
2. l2 is perpendicular to l1 if the slope of l2 is ________.
D. Tell whether the lines y = 2x – 8 and x + 2y = 5 are parallel, perpendicular or neither.
1. the slope of y = 2x – 8 is _______.
2. the slope of x + 2y = 5 is _______.
3. therefore the lines are ____________________.
E. Find the equation of the line parallel to the line x = 8 and containing (4, 9).
1. x = 8 is a __________________ line. A line parallel to it will also be vertical.
2. Therefore the equation of the parallel line will have the form x = a.
3. Since it contains (4, 9), the equation will be x = ______.
F. Find the equation of the line perpendicular to the line 2x + 3y = 9 and passing through the point (2, 6).
1. Putting 2x + 3y = 9 into slope-intercept form gives: y = (-2/3)x + _____
2. The slope of this line is __________.
3. Therefore the slope of a line perpendicular to this is 3/2.
4. Write the equation of the perpendicular line in point-slope form: y – _____ = 3/2(x – ____).
5. Putting into slope-intercept form gives: y = (3/2)x + _____.
IX. Using a Graphing Calculator to Make Calculations
A. If you have not used a graphing calculator before, you should familiarize yourself with some basic skills at this time. Go the David Williamson’s Graphing Calculator Tutorials at http://dtc.pima.edu/~dwilliamson/TI/indexti.html . The index of tutorials will appear on the left side of the screen. Click on one of these topics to view it. The topics you should view at this time are:
1. Second Function Key
2. Basic Calculations & Editing
3. Basic Calculations – Negative #s
4. Basic Calculations – Exponents
5. Basic Calculations – Roots
B. Use the graphing calculator to evaluate: -5+3^2.1-∛2 to the nearest thousandth. ____________
