+1 862 207 3288 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: ( ________ ) ∪ [ ________ ).
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