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 Graphs/156326: graph the inequality 2x+5y>5 1 solutions Answer 115133 by jim_thompson5910(28595)   on 2008-09-11 08:08:27 (Show Source): You can put this solution on YOUR website! Start with the given inequality. Subtract from both sides. Divide both sides by to isolate . Subtract from both sides. Rearrange the terms. Break up the fraction. Reduce So in order to graph , we need to graph . But first, we must graph the equation Now plug in the test point (0,0) into Since the inequality is false, this means that we shade the entire region that does NOT contain (0,0). So this means that we shade everything above the line Graph of where the boundary (which should be a dotted line) is the equation and the shaded region is in green.
Polynomials-and-rational-expressions/156339: 5. How do I complete this using factoring? Thank you.

x^2 + 6x + 5.
1 solutions

Answer 115130 by jim_thompson5910(28595)   on 2008-09-11 07:59:58 (Show Source):
You can put this solution on YOUR website!

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :
1,5
-1,-5

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .
1*5
(-1)*(-5)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First NumberSecond NumberSum
151+5=6
-1-5-1+(-5)=-6

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

---------------------------------------------

So factors to .

Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Polynomials-and-rational-expressions/156342: 8. Factoring is very hard for me. How do I complete this one?
15x^2 + 7x - 2
1 solutions

Answer 115129 by jim_thompson5910(28595)   on 2008-09-11 07:58:38 (Show Source):
You can put this solution on YOUR website!

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .
1*(-30)
2*(-15)
3*(-10)
5*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

---------------------------------------------

So factors to .

Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Polynomials-and-rational-expressions/156340: 6. How do I complete this using factoring? I appreciate the help.

8x^2 - 22x - 21
1 solutions

Answer 115128 by jim_thompson5910(28595)   on 2008-09-11 07:58:00 (Show Source):
You can put this solution on YOUR website!

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :
1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168
-1,-2,-3,-4,-6,-7,-8,-12,-14,-21,-24,-28,-42,-56,-84,-168

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .
1*(-168)
2*(-84)
3*(-56)
4*(-42)
6*(-28)
7*(-24)
8*(-21)
12*(-14)
(-1)*(168)
(-2)*(84)
(-3)*(56)
(-4)*(42)
(-6)*(28)
(-7)*(24)
(-8)*(21)
(-12)*(14)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First NumberSecond NumberSum
1-1681+(-168)=-167
2-842+(-84)=-82
3-563+(-56)=-53
4-424+(-42)=-38
6-286+(-28)=-22
7-247+(-24)=-17
8-218+(-21)=-13
12-1412+(-14)=-2
-1168-1+168=167
-284-2+84=82
-356-3+56=53
-442-4+42=38
-628-6+28=22
-724-7+24=17
-821-8+21=13
-1214-12+14=2

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

---------------------------------------------

So factors to .

Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

Graphs/156298: Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x.

5. f(x) = ln x

1 solutions

Answer 115099 by jim_thompson5910(28595)   on 2008-09-10 21:11:48 (Show Source):
You can put this solution on YOUR website!

# 5

Note: ln(x) also looks like LN(x) (to pronounce it, simply read off the letters "L" "N")

This is the natural log of x. So it is a logarithmic function.

Let's find the y value when

Plug in .

Take the natural log of 1 to get 0

So when , .

----------------------------

Let's find the y value when

Plug in .

Take the natural log of 2 to get 0.693

So when , .

----------------------------

Let's find the y value when

Plug in .

Take the natural log of 4 to get 1.386

So when , .

-------------------------------

Let's find the y value when

Plug in .

Take the natural log of 8 to get 2.079

So when , .

-------------------------------------

Let's find the y value when

Plug in .

Take the natural log of 16 to get 2.773

So when , .

Now let's make a table of the values we just found.

#### Table of Values:



xy10
20.693
41.386
82.079
162.773



Since the natural log function is logarithmic, this means that the growth is logarithmic. This growth rate is slower than linear growth rate and is the slowest growth rate than all of the growth rates.

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So here's the order of function's growth from smallest growth rate to largest growth rate

, , , and , and

Graphs/156297: Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x.

4. f(x) = 10^x

1 solutions

Answer 115098 by jim_thompson5910(28595)   on 2008-09-10 21:09:56 (Show Source):
You can put this solution on YOUR website!
# 4

Let's find the function value when :

Plug in .

Raise 10 to the first power to get 10

So if , then .

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Let's find the function value when :

Plug in .

Raise 10 to the second power to get 100

So if , then .

-------------

Let's find the function value when :

Plug in .

Raise 10 to the 4th power to get 10,000

So if , then .

-------------

Let's find the function value when :

Plug in .

Raise 10 to the 8th power to get 100,000,000

So if , then .

-------------

Let's find the function value when :

Plug in .

Raise 10 to the 16th power to get (this is a 1 followed by 16 zeros)

So if , then .

-------------

Now let's make a table of the values we just found.

#### Table of Values:



xy110
2100
410,000
8100,000,000
161*10^16



Since we are dealing with an exponential function, this means that the function undergoes exponential growth. This is the fastest of all of the growth rates in this group.

Graphs/156295: Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x.

3. f(x) = 2x3 + 7x2 - x - 1

1 solutions

Answer 115097 by jim_thompson5910(28595)   on 2008-09-10 21:02:33 (Show Source):
You can put this solution on YOUR website!
# 3

Let's find the function value when :

Plug in .

Cube to get .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Cube to get .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Cube to get .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Cube to get .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Cube to get .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Now let's make a table of the values we just found.

#### Table of Values:



xy17
241
4235
81463
169967



Since this polynomial is a cubic, this tells us that the rate of growth is cubic growth. This growth rate is larger than quadratic growth.

Graphs/156294: Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x.
2. f(x) = x2 - 3x + 2

1 solutions

Answer 115096 by jim_thompson5910(28595)   on 2008-09-10 21:01:06 (Show Source):
You can put this solution on YOUR website!
# 2

Let's find the function value when :

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Now let's make a table of the values we just found.

#### Table of Values:



xy10
20
46
842
16210



Because the function is a quadratic, this means that the polynomial undergoes quadratic growth. This is faster than linear growth.

Graphs/156293: Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x.
1. f(x) = 5x - 3

1 solutions

Answer 115095 by jim_thompson5910(28595)   on 2008-09-10 20:59:27 (Show Source):
You can put this solution on YOUR website!
# 1

Let's find the function value when :

Plug in .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Let's find the function value when :

Plug in .

Multiply and to get .

Combine like terms.

So if , then .

-------------

Now let's make a table of the values we just found.

#### Table of Values:



xy12
27
417
837
1677



Since this is a linear equation, this means that the equation experiences linear growth. This is considered moderate growth.

 Graphs/156282: 3. Plot the graph of the above equations formed in question 2, and post your response to the discussion forum. 1 solutions Answer 115092 by jim_thompson5910(28595)   on 2008-09-10 20:27:47 (Show Source): You can put this solution on YOUR website! Here are the graphs to question 2 a) Graph of b) Graph of c) Graph of c) Graph of d) Graph of e) Graph of
 Graphs/156280: 2. Form each of the following: • A linear equation in one variable • A linear equation in two variables • A quadratic equation • A polynomial of three terms • An exponential function • A logarithmic function 1 solutions Answer 115091 by jim_thompson5910(28595)   on 2008-09-10 20:27:00 (Show Source): You can put this solution on YOUR website! a) A linear equation in one variable : b) A linear equation in two variables : c) Quadratic equation: d) A polynomial of three terms: e) Exponential Function: f) Logarithmic Function:
 Graphs/156278: 1. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph of both the functions and post to the discussion forum. Discuss these functions and their graphs with your classmates1 solutions Answer 115088 by jim_thompson5910(28595)   on 2008-09-10 20:24:33 (Show Source): You can put this solution on YOUR website!Example of an Exponential function: Now convert the Exponential function to a Logarithmic function using the property <====> : Now let's graph the exponential function (red) and the logarithmic function (green) Notice how the logarithmic function is simply a reflection of the exponential function over the line
 Inequalities/156129: 2(5y-6) > (3y+6) - Solution is {y|y __ __ }1 solutions Answer 115041 by jim_thompson5910(28595)   on 2008-09-10 14:17:11 (Show Source): You can put this solution on YOUR website! Start with the given inequality. Distribute. Add to both sides. Subtract from both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . ---------------------------------------------------------------------- Answer: So the answer is Which approximates to So the solution set is Also, the answer in interval notation is [) Finally, here's the graph of the solution set Note: the endpoint is an closed circle
 Expressions-with-variables/156164: simplify 6-a+2ac+5c-3ca1 solutions Answer 115034 by jim_thompson5910(28595)   on 2008-09-10 14:04:19 (Show Source): You can put this solution on YOUR website! Start with the given expression Group the common terms. Note Combine like terms So
 Geometry_Word_Problems/156165: Hello; I was wondering if you could help me with the following problem; What is the equation of the line that contains the points with (x,y) coordinates (-3,7) and (5,-1)? Sorry, I don't have book's name or the number.1 solutions Answer 115033 by jim_thompson5910(28595)   on 2008-09-10 13:54:42 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Reduce So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Rewrite as Distribute Multiply Add 7 to both sides. Combine like terms. Simplify So the equation that goes through the points and is Notice how the graph of goes through the points and . So this visually verifies our answer. Graph of through the points and
 Polynomials-and-rational-expressions/156166: CAN SOMEONE PLEASE HELP ME WITH THIS PROBLEM: If f(g(x)) = g(f(x)) = x, then whay can we say about f(x) and g(x)? a. they are functional inverses of each other b. f(x) = g(x) c. f(x) = g(x) = x d. nothing can be said about f(x) and g(x) 1 solutions Answer 115031 by jim_thompson5910(28595)   on 2008-09-10 13:53:52 (Show Source): You can put this solution on YOUR website!Remember, if f(x) and g(x) are inverses of one another, then we can say that f(g(x))=x and g(f(x))=x. So this means that the answer is a) they are functional inverses of each other Note: we know nothing about f(x) and g(x). So we cannot just blindly assume that f(x)=g(x) or f(x)=g(x)=x without some evidence. So this rules our choices b) and c).
 Functions/156168: I have one more problem that hopefully you'll be able to help me; If f(4)=0 and f(6) =6, which of the following could represent f(x)? A. 2/3x-4 B. x+2 C. x-4 D. 3/2x+6 E. 3x-121 solutions Answer 115030 by jim_thompson5910(28595)   on 2008-09-10 13:50:32 (Show Source): You can put this solution on YOUR website!The statement tells us that if , then . So the point (4,0) is on the line. Also, the statement tells us that if , then . So the point (6,6) is also on the line. So let's find the equation of the line that goes through the two points (4,0) and (6,6) To do that, we first need to find the slope of the line through the points and Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Reduce So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Distribute Multiply Add 0 to both sides. Combine like terms. Simplify So the equation that goes through the points and is In function notation, the answer is Notice how and . So this also verifies our answer. Also, notice how the graph of goes through the points and . So this visually verifies our answer. Graph of through the points and
 Polynomials-and-rational-expressions/156161: CAN SOMEONE PLEASE HELP ME WITH THIS PROBLEM: What is g(x) = f^-1(x) if f(x) = 1/2 x? a. g(x) = 0 b. g(x) = -2x c. g(x) = 1/2x^-1 d. g(x) = 2x1 solutions Answer 115028 by jim_thompson5910(28595)   on 2008-09-10 13:30:59 (Show Source): You can put this solution on YOUR website!Is the function ? Start with the given expression Replace with "y" Switch "x" and "y" Multiply both sides by 2 to solve for "y" So after solving for "y", we get . So the inverse function is ====================================== Or... Is the function ? Start with the given expression Replace with "y" Switch "x" and "y" Multiply both sides by Divide both sides by to solve for "y" So after solving for "y", we get . So the inverse function is
 Quadratic_Equations/156147: I have a swan that weighs 25 lbs. and is 4 ft. long. Using this equation what is the L in wing span in feet. Equation: L=2.43*W^0.33261 solutions Answer 115027 by jim_thompson5910(28595)   on 2008-09-10 13:26:39 (Show Source): You can put this solution on YOUR website!I think they threw in the fact that the swan "is 4 ft. long" to throw you off (since it's extra info that you don't need) Start with the given equation Plug in Raise 25 to the 0.3326th power to get 2.917 Multiply 2.43 and 2.917 to get 7.08831 So the approximate wingspan (we introduced rounding errors) is about 7.09 feet (rounded to the nearest hundredth).
 Matrices-and-determiminant/156056: You have been performing Gaussian Elimination, and now you have all of the lower triangle elements with value 0. What are you ready to do? I think the answer is c, but I want to make sure a. Nothing, you are done with Gaussian Elimination b. Start with forwards elimination c. Start backwards elimination d. Start over; you are supposed to have all zeroes in the upper triangle1 solutions Answer 115025 by jim_thompson5910(28595)   on 2008-09-10 13:21:17 (Show Source): You can put this solution on YOUR website!You are correct. You want to end up with a matrix with a diagonal of 1's where every other element (except for the far right column) is zero. Note: It's not always possible to obtain this since you may have no solutions or an infinite number of solutions.
 Expressions-with-variables/156158: Expand 5y{x-2y+3}1 solutions Answer 115022 by jim_thompson5910(28595)   on 2008-09-10 13:17:39 (Show Source):
 Matrices-and-determiminant/156047: If A is a 3 × 4 matrix, B is a 4 × 12 matrix, and C is a 12 × 2 matrix, then CBA is: a. 13,824 b. a 3 × 4 × 12 matrix c. a 3 × 2 matrix d. none of the above, these matrices do not match up1 solutions Answer 115021 by jim_thompson5910(28595)   on 2008-09-10 13:16:41 (Show Source): You can put this solution on YOUR website!Notice how B is 4 × 12 and A is 3 × 4 The number of columns in matrix B (12) do NOT equal the number of rows in matrix A (3). So this means that BA is not possible. So the answer choice is d) none of the above, these matrices do not match up
 expressions/156007: 5(2m+5)-6 1 solutions Answer 115020 by jim_thompson5910(28595)   on 2008-09-10 13:13:09 (Show Source):
Square-cubic-other-roots/156155: Is 4 or -1 a root of the equation ysquared-3y=4?
1 solutions

Answer 115017 by jim_thompson5910(28595)   on 2008-09-10 13:10:55 (Show Source):
You can put this solution on YOUR website!

Subtract 4 from both sides

 Solved by pluggable solver: Quadratic Formula Let's use the quadratic formula to solve for y: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=1, b=-3, and c=-4 Negate -3 to get 3 Square -3 to get 9 (note: remember when you square -3, you must square the negative as well. This is because .) Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 1 to get 2 So now the expression breaks down into two parts or Lets look at the first part: Add the terms in the numerator Divide So one answer is Now lets look at the second part: Subtract the terms in the numerator Divide So another answer is So our solutions are: or

So 4 or -1 is a root of the equation

 Trigonometry-basics/156050: Adding together the first ten natural numbers is an example of a(n): a. series b. arithmetic series c. geometric series d. power series1 solutions Answer 115016 by jim_thompson5910(28595)   on 2008-09-10 13:09:12 (Show Source): You can put this solution on YOUR website!Adding together the first ten natural numbers is an example of a(n): b. arithmetic series An "arithmetic sequence" is simply a sequence where you add a constant number to each term. In this case, you add 1 to each term. An "arithmetic series" is simply the sequence of sums of the arithmetic sequence.
 Polynomials-and-rational-expressions/156142: If x^2+6x+16=(x+a)^2+b for all values of x. Find a and b1 solutions Answer 115015 by jim_thompson5910(28595)   on 2008-09-10 13:07:14 (Show Source): You can put this solution on YOUR website! Start with the given equation FOIL Subtract from both sides Notice how the term is the only term on the right side that has an "x" in it. So this means that Start with the given equation Divide both sides by to isolate "a" So the first answer is ------------------------- Looking back at , the terms do not have an "x" term in them at all. So this means that Start with the given equation Plug in a=3 Square 3 to get 9 Subtract 9 from both sides So the second answer is Check: } works
15x^4-31x^2+10=0
The solution of x= ??
1 solutions

Answer 115014 by jim_thompson5910(28595)   on 2008-09-10 13:01:22 (Show Source):
You can put this solution on YOUR website!
Let . So . In short,

Plug in and

 Solved by pluggable solver: Quadratic Formula Let's use the quadratic formula to solve for z: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=15, b=-31, and c=10 Negate -31 to get 31 Square -31 to get 961 (note: remember when you square -31, you must square the negative as well. This is because .) Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 15 to get 30 So now the expression breaks down into two parts or Lets look at the first part: Add the terms in the numerator Divide So one answer is Now lets look at the second part: Subtract the terms in the numerator Divide So another answer is So our solutions are: or

Remember, we let . So this means that

or

Take the square root of both sides for each case (remember the "plus/minus")

, , , or

Rationalize the denominator (if necessary)

, , , or

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