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 Functions/252220: What is the range of the function y = |2x - 3| - 5?1 solutions Answer 184046 by jim_thompson5910(28476)   on 2009-12-25 00:55:09 (Show Source): You can put this solution on YOUR website!If you plot , you get: From the graph, we see that the lowest point is (1.5,-5). So the smallest value in the range is -5 (since this is the smallest y value that is on the graph). This means that the range is which looks like [) in interval notation.
 Equations/252214: y varies inversely as x. Write the appropriate inverse-variation equation, and find y for the given values of x. y=14 when x=8; x=values:10, 15, and 201 solutions Answer 184045 by jim_thompson5910(28476)   on 2009-12-25 00:27:02 (Show Source): You can put this solution on YOUR website!Since "y varies inversely as x", this means that . Because "y=14 when x=8", this means that we can plug those values in to get: . Multiply both sides by 8 to isolate k to get: . So the constant k is So the inverse-variation equation is To find the y values when x=10, 15, and 20, simply plug these values into 'x' of the equation and simplify. Ex: When x=10, then . So if , then I'll let you do the rest.
 Functions/252218: Let W be all the words in English. A relation L is defined from W to the positive integers that associates each word with the number of letters in the word. For example (L)the = 3 since the word the has three letters. (a) What pairs in L have second coordinate 1? (b) Is this relation a function? Explain your answer. For (b) I do not believe it is, but that's just a wild guess from the little knowledge I know about this topic. Anyone's help on this would be greatly appreciated! -P.M.1 solutions Answer 184043 by jim_thompson5910(28476)   on 2009-12-25 00:21:10 (Show Source): You can put this solution on YOUR website!a) Just pick a word that has a length of 1. So any letter will do. b) Since each word has a specific length (and only ONE specific length), this means that the relation L is indeed a function. If L mapped say the word 'the' to the numbers 3 AND 4 (say hypothetically that 'the' was 3 characters and 4 characters long), then L wouldn't be a function since that input yielded two outputs. So in general, if every input into the relation yields only one output, then that relation is a function.
Polynomials-and-rational-expressions/252211: u^2-2uv-63v^2
----------------
x^2+2xy-24y^2
----------------
u^2-2uv-8v^2
----------------
u^2-2uv-8v^2
----------------
u^2-3uv-40v^2
----------------
x^2+50x+625
----------------
x^2-8xy+16y^2
1 solutions

Answer 184036 by jim_thompson5910(28476)   on 2009-12-24 22:47:23 (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started.

# 1

Looking at we can see that the first term is and the last term is where the coefficients are 1 and -63 respectively.

Now multiply the first coefficient 1 and the last coefficient -63 to get -63. Now what two numbers multiply to -63 and add to the middle coefficient -2? Let's list all of the factors of -63:

Factors of -63:
1,3,7,9,21,63

-1,-3,-7,-9,-21,-63 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -63
(1)*(-63)
(3)*(-21)
(7)*(-9)
(-1)*(63)
(-3)*(21)
(-7)*(9)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First NumberSecond NumberSum
1-631+(-63)=-62
3-213+(-21)=-18
7-97+(-9)=-2
-163-1+63=62
-321-3+21=18
-79-7+9=2

From this list we can see that 7 and -9 add up to -2 and multiply to -63

Now looking at the expression , replace with (notice combines back to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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

So factors to

=====================================================================

# 2

Looking at we can see that the first term is and the last term is where the coefficients are 1 and -24 respectively.

Now multiply the first coefficient 1 and the last coefficient -24 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient 2? Let's list all of the factors of -24:

Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2

From this list we can see that -4 and 6 add up to 2 and multiply to -24

Now looking at the expression , replace with (notice combines back to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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

So factors to

The rest of the problems are all similar to these two. So I'll let you finish up.

 Equations/252209: 30-7z=10z-41 solutions Answer 184006 by jim_thompson5910(28476)   on 2009-12-24 18:33:33 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract from 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 . Reduce. ---------------------------------------------------------------------- Answer: So the solution is
 Equations/252158: Can you tell me the steps to solving this problem... -3(2x-5)=7-4(x-6)1 solutions Answer 183940 by jim_thompson5910(28476)   on 2009-12-23 22:46:34 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. Combine like terms on the right side. Subtract from both sides. Add to both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is
 Polynomials-and-rational-expressions/249902: x^3+6x^2+7x+421 solutions Answer 183939 by jim_thompson5910(28476)   on 2009-12-23 20:05:21 (Show Source): You can put this solution on YOUR website!I'm assuming you want to factor this right? Start with the given expression Group like terms Factor out the GCF out of the first group. Factor out the GCF out of the second group Since we have the common term , we can combine like terms So factors to In other words,
 Exponents/252151: The assignment is as follows: Look at the expressions below. Write a narrative description of what you would do to simplify each one. 1. (16)^(1/2) 2. √100 I struggle with these kind of assignments. Im not sure how to interpret the assignment. My first thought is the answer is simply I would calculate the square root of each number. That is what I would do to simplify each one. Does anyone have any thoughts on what the instructor is looking for? Thanks! 1 solutions Answer 183931 by jim_thompson5910(28476)   on 2009-12-23 17:33:22 (Show Source): You can put this solution on YOUR website!Since , you are correct in saying that you are going to take the square root of each number. One way you can do it is to write 16 as and 100 as and then realize that (where . So this means that .
 Permutations/252150: Suppose you are playing a word game with seven distinct letters. How many seven-letter words can there be?1 solutions Answer 183929 by jim_thompson5910(28476)   on 2009-12-23 17:20:45 (Show Source): You can put this solution on YOUR website!Assuming that the letters can only be used once, you have 7 letters for the first slot, 6 letters for the second (since one has been used already), 5 for the third, etc. until you have one left over for the last slot. Multiply these values to get: 7*6*5*4*3*2*1 = 5040 So there are 5,040 different seven-letter 'words'. Note: a 'word' in this case is just a string of letters (it may or may not be a real word)
 Polynomials-and-rational-expressions/252147: Show that 2x^2-12x+23 is equal to 2(x+3)^2+51 solutions Answer 183928 by jim_thompson5910(28476)   on 2009-12-23 16:28:11 (Show Source): You can put this solution on YOUR website!I think you made a typo. The problem should be "Show that 2x^2-12x+23 is equal to 2(x-3)^2+5" Start with the given expression. Factor out the coefficient . This step is very important: the coefficient must be equal to 1. Take half of the coefficient to get . In other words, . Now square to get . In other words, Now add and subtract inside the parenthesis. Make sure to place this after the "x" term. Notice how . So the expression is not changed. Group the first three terms. Factor to get . Combine like terms. Distribute. Multiply. So after completing the square, transforms to . So for all real values of 'x'.
 Parallelograms/252102: A parallelogram has sides of length 36.4 cm and 21.5 cm. If the longer diagonal has a length of 38.9 cm, what is the angle opposite this diagonal? I have tried drawing a picture and setting up an equation but I cant figure it out. Thanks 1 solutions Answer 183860 by jim_thompson5910(28476)   on 2009-12-23 00:14:02 (Show Source): You can put this solution on YOUR website!Hint: use the law of cosines where 'a', 'b', and 'c' are the lengths of the triangle and 'A' is the angle opposite the side 'a'. So this problem will make and you'll let 'b' and 'c' be equal to the remaining values given. Let me know if this helps at all.
 Equations/252085: 7s + 4(3 - s) = 181 solutions Answer 183838 by jim_thompson5910(28476)   on 2009-12-22 19:38:48 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. Combine like terms on the left side. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is
 Complex_Numbers/252083: please solve for Y (X-Y)/Y=Z I got as far as: 1/Y=Z/X + 1/X1 solutions Answer 183836 by jim_thompson5910(28476)   on 2009-12-22 19:36:54 (Show Source): You can put this solution on YOUR website! Start with the given equation. Multiply both sides by 'y'. Add 'y' to both sides. Factor out the GCF 'y' from the right side. Divide both sides by to isolate 'y'. So the solution is Here's the confirmation of the answer.
 Equations/251912: What is the solution for v-4(4-v)=-2(2v-1)?1 solutions Answer 183612 by jim_thompson5910(28476)   on 2009-12-21 15:19:22 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. Combine like terms on the left side. Add to both sides. Add to both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is
 Triangles/251891: Happy holidays. I am having trouble with this question and i think it should be in this section. Please help me. I am so lost. heres the link http://www2.edmastery.com/li/itemAssets/NN.MAT.301307.1.HO%2Fres00016%2F21fc5023cb6e448e87f4efd448374370%2FUnit3question2.jpg1 solutions Answer 183592 by jim_thompson5910(28476)   on 2009-12-21 13:42:27 (Show Source): You can put this solution on YOUR website!It's not stated, but there is another fact about this figure that HK = HK. Since IK = JH and HI = KJ also, we have 3 pairs of congruent sides which means that SSS will be used in this case.
 Geometry_proofs/251890: Hello. i am having trouble with this proof and im hoping that someone will be able to help me. i believe that the answer is the angle side angle theorem but im not quite sure. help? heres the link http://www2.edmastery.com/li/itemAssets/NN.MAT.301307.1.HO%2Fres00016%2Fbdeacaa642ea4394ad6b31e45cf605d3%2FUnit3question.jpg1 solutions Answer 183591 by jim_thompson5910(28476)   on 2009-12-21 13:38:18 (Show Source): You can put this solution on YOUR website!Since you're given that AB = CD, AD = CB, and BD = BD, this means that you are basically showing that these three pairs of sides are congruent to one another. So this would then mean that the SSS (Side Side Side) Property of Congruence would be used.
 Linear-equations/251888: Life expectancies at birth for men and women can be modeled by the following functions: w(x)= 0.126x+76.74 M(x)= 0.126x+69.11 W(x) represents the life expectancy of women and M(x) represents the life expectancy of men. The x represents the years since 1975. (x=0 corresponds to the year 1975, x=5 corresponds to 1980, and so on). this goes on until 2000 x is the years since 1975. 1975 0 1985-5 1990-15 1995- 20 2000- 25 I am not understanding this I also have to figure out on a slope is the life expectantcy of men or women increasing more rapidly. then I have to determine the birth years in a inequality where the men life expectancy are greater than women 1 solutions Answer 183590 by jim_thompson5910(28476)   on 2009-12-21 13:33:46 (Show Source): You can put this solution on YOUR website!Since the equation is in slope-intercept form , where 'm' is the slope and 'b' is the y-intercept, this means that the slope is . Similarly, for , this equation is also in slope-intercept form and the slope is as well. So the two slopes are equal. What this means is that the life expectancies are both increasing at the same rate. To find out which years men have a greater life expectancy than women, simply set M(x) (life expectancy of men) greater than W(x) (life expectancy of women) like this Plug in and Multiply both sides by 1000 to clear out the decimals. Distribute and multiply. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Combine like terms on the right side. Simplify. Since this inequality is NEVER true for any value of 'x', this means that the original inequality is also NEVER true. Since there are no solutions to the original inequality, this means that the life expectancy of men will never exceed the life expectancy of women. Note: I would make sure that you copied the problem down correctly (or ask if there are any typos).
 Square-cubic-other-roots/251887: Did I do this equation right? 3 sqrt of 12 + 2 sqrt of 3 3 sqrt (4*3) + 2 sqrt 3 3+2+2 sqrt 3 7 sqrt 3 Is this correct?? thanks :) 1 solutions Answer 183583 by jim_thompson5910(28476)   on 2009-12-21 13:12:15 (Show Source): You can put this solution on YOUR website! Start with the given expression Simplify to get . Note: If you need help with simplifying square roots, check out this solver. Multiply 3 and 2 to get 6. Since we have the common term , we can combine like terms Factor out the GCF Combine like terms. So simplifies to . In other words,
 Equations/251828: Can somebody help me find x: 1 solutions Answer 183525 by jim_thompson5910(28476)   on 2009-12-20 23:24:41 (Show Source): You can put this solution on YOUR website!Take note that . So . Also, keep in mind that Start with the given equation. Square the individual terms (see above). Combine like terms (this just amounts to adding fractions). Multiply both sides by 4. Divide both sides by 5. Cancel out the common terms. Simplify. Take the square root of both sides. or Break up the "plus/minus" to form two equations. or Take the square root of to get . or Combine like terms. -------------------------------------- Answer: So the solutions are or .
 Radicals/251829: x^2+2x+9 I have tried x={-2ħsqrt(2^2-4*1*9)}/2*1 (quadratic formula: x={-bħsqrt( b^2-4ac)}/2(a)) Process: x={-2ħsqrt(2^2-4*1*9)}/2(1) x={-2ħsqrt(4-36)}/2 x=(-2/2)ħ{sqrt-32/2} x=-1ħ(sqrt-32/2) Looking for -1ħ2i*sqrt(2), but I can't get there. I need the process to find this imaginary root. what am I doing wrong?1 solutions Answer 183524 by jim_thompson5910(28476)   on 2009-12-20 23:20:16 (Show Source): You can put this solution on YOUR website!The trick here lies in simplifying Start with the given expression Factor out a negative 1 Break up the square roots using the identity Replace with (remember ) Now lets simplify : The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number. So let's list the factors of 32 Factors: 1, 2, 4, 8, 16, 32 Notice how 16 is the largest perfect square, so lets factor 32 into 16*2 Factor 32 into 16*2 Break up the square roots using the identity Take the square root of the perfect square 16 to get 4 This means that the expression simplifies to So the expression simplifies to (just reintroduce back in) In other words, ---------------------------------------------------------------------- Going back to , this expression simplifies to . From there, it reduces to So the solutions are or
 Equations/251816: Solve for the variable. 2(X-1)+3X=6X-20 keainc74@aol.com1 solutions Answer 183498 by jim_thompson5910(28476)   on 2009-12-20 21:26:33 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. Combine like terms on the left side. 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 . Reduce. ---------------------------------------------------------------------- Answer: So the solution is
 Miscellaneous_Word_Problems/251805: Hockey teams recieve 2 points when they win and 1 point when they tie. One season, a team won a championship with 60 points. They won 9 more games than they tied. How many wins and how many ties did the team have? How many wins did the team have? How many ties did the team have?1 solutions Answer 183494 by jim_thompson5910(28476)   on 2009-12-20 20:13:32 (Show Source): You can put this solution on YOUR website!Check out this solution.
 Rectangles/251804: if the perimeter of a rectangle is expressed by the formula p=2l + 2w, the side w can be found by what formula1 solutions Answer 183493 by jim_thompson5910(28476)   on 2009-12-20 20:07:05 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract 2L from both sides. Divide both sides by 2 to isolate W. So the answer is
 Miscellaneous_Word_Problems/251800: For the following right angle, find the side length x round to the nearest tenth the numbers are x hypotunuse 12 side 10 side1 solutions Answer 183488 by jim_thompson5910(28476)   on 2009-12-20 19:54:00 (Show Source): You can put this solution on YOUR website! We basically have this triangle set up: To find the unknown length, we need to use the Pythagorean Theorem. Remember, the Pythagorean Theorem is where "a" and "b" are the legs of a triangle and "c" is the hypotenuse. Since the legs are and this means that and Also, since the hypotenuse is , this means that . Start with the Pythagorean theorem. Plug in , , Square to get . Square to get . Combine like terms. Rearrange the equation. Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense). Simplify the square root. Approximate the right side with a calculator. ================================================================ Answer: So the solution is approximately which means that the hypotenuse is roughly 15.6 (rounded to the nearest tenth) units long.
 Miscellaneous_Word_Problems/251801: If the expression is to make sense 5w-6/7w+8 the values that w can take must be restricted, state the restriction on w1 solutions Answer 183485 by jim_thompson5910(28476)   on 2009-12-20 19:51:37 (Show Source): You can put this solution on YOUR website!Remember that you CANNOT divide by zero. So set the denominator equal to zero to get . Solve for 'w' to get . So when , the denominator is equal to zero which is what we don't want. This means that we must make the restriction that for the expression to make sense.
 Miscellaneous_Word_Problems/251799: yolanda runs 4 miles in 30 minutes, at the same rate how many miles would she run in 72 minutes1 solutions Answer 183483 by jim_thompson5910(28476)   on 2009-12-20 19:49:23 (Show Source): You can put this solution on YOUR website!Simply set up the equation where 'x' is the number of miles she runs in 72 minutes. Notice how the ratio of 4 to 30 is equal to the ratio of x to 72. Start with the given ratio Multiply both sides by 72 Multiply Reduce. So our answer is which in decimal form is which means that she can run 9.6 miles in 72 minutes.
 Polynomials-and-rational-expressions/251796: Please simplify: (x+y)(x^2+2xy-y)^2 1 solutions Answer 183480 by jim_thompson5910(28476)   on 2009-12-20 19:37:19 (Show Source): You can put this solution on YOUR website!First, let's expand Start with the given expression. Rewrite as Expand. Recall that Distribute. Multiply. Now combine like terms. So expands to . In other words, . ------------------------------------------------------------------------------ This means that then becomes Now let's finish up... Start with the given expression. Expand. Distribute. Multiply. Now combine like terms. So expands to . In other words, . ========================================================================== Answer: So fully expands out to In other words, Note: if this problem was more complicated than it should have been, then maybe there's a typo somewhere.
 Polynomials-and-rational-expressions/251794: 3x^2y^6z^3 + 2x^5y^3z^7 - 77 + 3x^5 what would the degree on this polynomial be?1 solutions Answer 183476 by jim_thompson5910(28476)   on 2009-12-20 19:25:38 (Show Source): You can put this solution on YOUR website!The degree of any monomial is just the sum of all of the exponents. So the degree of is and the degree of is . The degrees of 7 and are 0 and 5 respectively. The degree of the entire polynomial is the largest degree of all of the monomials present. So the degree of is 15 since this is the largest degree among the individual monomials.
 Linear_Algebra/251793: what is the x intercept of the equation 1 solutions Answer 183475 by jim_thompson5910(28476)   on 2009-12-20 19:23:03 (Show Source): You can put this solution on YOUR website! Start with the given equation. Plug in Simplify Divide both sides by -6. Simplify So when , giving us the point which is the x-intercept.