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 Sequences-and-series/199680: Find the x values so that the infinite series become finite. (infinite) (Sigma) 3(x-2)^(k-1) (k=1) Could you help me with this question? Thank you.1 solutions Answer 150060 by jim_thompson5910(28594)   on 2009-06-10 18:58:21 (Show Source): You can put this solution on YOUR website!The series will converge if and only if So this means that you simply need to solve to find the set of x values (since in this case, )
 Probability-and-statistics/199673: This is my last question please please help I'm not good in math with word problems. A couple plan to have exactly four children. (a) Construct a tree diagram and list the sample space. (b) Find the probability that the family has at least three boys. 1 solutions Answer 150059 by jim_thompson5910(28594)   on 2009-06-10 18:43:06 (Show Source): You can put this solution on YOUR website!a) Let's build the tree: Using the tree, we get the sample space: {b,b,b,b} {b,b,b,g} {b,b,g,b} {b,b,g,g} {b,g,b,b} {b,g,b,g} {b,g,g,b} {b,g,g,g} {g,b,b,b} {g,b,b,g} {g,b,g,b} {g,b,g,g} {g,g,b,b} {g,g,b,g} {g,g,g,b} {g,g,g,g} So for instance {b,g,g,b} means that the couple had a boy, girl, girl, and then a boy (in that order). Note: recall, the sample space is the set of ALL possible outcomes. ------------------------------- b) Since we want to know the chances of the couple having "at least three boys", this means they want to know the chances of having 3 boys OR 4 boys (since at least means that figure or more). Looking back at the list of all possible outcomes (ie the sample space) from part a), we see that we have the combinations for 3 boys: {b,b,b,g}, {b,b,g,b}, {b,g,b,b}, and {g,b,b,b} So there are 4 cases where the couple would have 3 boys. So ----------------------------- Also, since there is only ONE way to have 4 boys (of a total of 4 children), this means that ------------------------------------------------------- Now simply add the two probabilities to find the chances of either one occurring: So the probability of the couple having AT LEAST 3 boys is which is 0.3125 in decimal form which gives a 31.25% chance.
 Exponents/199666: Please help me answer this: 3^(4x) = 27^(x-2) Thanks.1 solutions Answer 150057 by jim_thompson5910(28594)   on 2009-06-10 18:22:32 (Show Source): You can put this solution on YOUR website! Start with the given equation. Rewrite 27 as Multiply the exponents. Since the bases are equal, the exponents are equal. Distribute. Subtract from both sides. Combine like terms on the left side. ---------------------------------------------------------------------- Answer: So the solution is
 Equations/199678: Solve the value of x in the equation below: 2(3x+1)=10x-61 solutions Answer 150056 by jim_thompson5910(28594)   on 2009-06-10 18:19:55 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. 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
 logarithm/199628: what values of x cannot possibly be solutions of the following equation? LOGa(3x+1)=21 solutions Answer 150020 by jim_thompson5910(28594)   on 2009-06-10 13:44:35 (Show Source): You can put this solution on YOUR website!Recall that you CANNOT evaluate the log of a negative value or 0. So this means that Start with the given inequality. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . This means that any values of x that are either equal to or are less than are NOT allowed.
 Linear-systems/199621: The difference between two numbers is 18. If the larger number is 5 more than twice the smaller number, what is their sum?1 solutions Answer 150018 by jim_thompson5910(28594)   on 2009-06-10 13:35:30 (Show Source): You can put this solution on YOUR website!"The difference between two numbers is 18" translates to "If the larger number is 5 more than twice the smaller number" translates to Start with the first equation. Plug in Combine like terms on the left side. Subtract from both sides. Combine like terms on the right side. Go back to the second equation. Plug in Multiply Add Now simply add up x and y to get: So
 logarithm/199620: Solve 3^x-1= 9^2x1 solutions Answer 150017 by jim_thompson5910(28594)   on 2009-06-10 13:31:18 (Show Source): You can put this solution on YOUR website! Start with the given equation. Rewrite 9 as Multiply the exponents Multiply Since the bases are equal, the exponents are equal. Add to both sides. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is which approximates to .
 absolute-value/199616: name two different numbers that have an absolute value of 71 solutions Answer 150016 by jim_thompson5910(28594)   on 2009-06-10 12:52:45 (Show Source): You can put this solution on YOUR website!Let x=unknown number What you want to solve is Break up the absolute value to get or Note: if , then or ) So the solutions are or which means that the absolute value of -7 or 7 is 7.
Quadratic_Equations/199600: I'm not to sure if this is the right section for my question but here goes:
I'm working on factorizing quadratic trinomials and here is a question I need help with factorizing this expression.
1) 20x^2-44x+21
1 solutions

Answer 150015 by jim_thompson5910(28594)   on 2009-06-10 12:32:40 (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 :
1,2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420
-1,-2,-3,-4,-5,-6,-7,-10,-12,-14,-15,-20,-21,-28,-30,-35,-42,-60,-70,-84,-105,-140,-210,-420

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

These factors pair up and multiply to .
1*420
2*210
3*140
4*105
5*84
6*70
7*60
10*42
12*35
14*30
15*28
20*21
(-1)*(-420)
(-2)*(-210)
(-3)*(-140)
(-4)*(-105)
(-5)*(-84)
(-6)*(-70)
(-7)*(-60)
(-10)*(-42)
(-12)*(-35)
(-14)*(-30)
(-15)*(-28)
(-20)*(-21)

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

First NumberSecond NumberSum
14201+420=421
22102+210=212
31403+140=143
41054+105=109
5845+84=89
6706+70=76
7607+60=67
104210+42=52
123512+35=47
143014+30=44
152815+28=43
202120+21=41
-1-420-1+(-420)=-421
-2-210-2+(-210)=-212
-3-140-3+(-140)=-143
-4-105-4+(-105)=-109
-5-84-5+(-84)=-89
-6-70-6+(-70)=-76
-7-60-7+(-60)=-67
-10-42-10+(-42)=-52
-12-35-12+(-35)=-47
-14-30-14+(-30)=-44
-15-28-15+(-28)=-43
-20-21-20+(-21)=-41

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).

 Rational-functions/199584: b^2-9/b^3-27 i know i have to use the cube root but i stilll dont get how this problem is sloved.. can you please explain this (:1 solutions Answer 150010 by jim_thompson5910(28594)   on 2009-06-10 12:09:37 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get (use the difference of squares formula) Factor to get (use the difference of cubes formula) Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where
 Equations/199604: 3M+20=2M+801 solutions Answer 150006 by jim_thompson5910(28594)   on 2009-06-10 12:01:55 (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. ---------------------------------------------------------------------- Answer: So the solution is
Matrices-and-determiminant/199607: Hello,
I'm trying to find the inverse of the following matrix; please help. I have to show all my work for homework, but I'm getting confused. Thank you!
(18 9)
(-8 -4)
1 solutions

Answer 150004 by jim_thompson5910(28594)   on 2009-06-10 12:00:56 (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Finding the Inverse of a 2x2 Matrix

To find the inverse of the matrix , we can follow these steps:

### Step 1) Find the determinant

The determinant of is . So this means that

Since the determinant is equal to zero, this means that no inverse exists.

Remember, the inverse of is .

If , then a division by zero occurs, which is NOT allowed.

So we can stop here.

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

So the inverse of does NOT exist since (ie the determinant is 0).

 Linear-equations/199610: This question is from textbook intermediate algebra 3x-y=-3 2x+y=-7 i am trying to figure out how to find the x,y,xy corrdinates so i can graph the equation to find out this problem has one solution, no solution, or infininate solutions1 solutions Answer 150002 by jim_thompson5910(28594)   on 2009-06-10 11:59:10 (Show Source): You can put this solution on YOUR website!Start with the given system of equations: In order to graph these equations, we must solve for y first. Let's graph the first equation: Start with the first equation. Subtract from both sides. Divide both sides by to isolate . Rearrange the terms and simplify. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 3 and the run is 1. This means that to go from point to point, we can go up 3 and over 1 So starting at , go up 3 units and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and ---------------------------------------------------------------------------------------- Now let's graph the second equation: Start with the second equation. Subtract from both sides. Rearrange the terms and simplify. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is -2 and the run is 1. This means that to go from point to point, we can go down 2 and over 1 So starting at , go down 2 units and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and ---------------------------------------------------------------------------------------- Now let's graph the two equations together: Graph of (red). Graph of (green) From the graph, we can see that the two lines intersect at the point . So the solution to the system of equations is . This means that the system has one unique solution. This also tells us that the system of equations is consistent and independent.
Polynomials-and-rational-expressions/199608: This question is from textbook Beginning and Intermediate Algebra
Find any variables for which the rational expression is undefined.
5m + 2
_________
m(squared) + m - 6
I tried to set the denominator equal to 0, then factor, but I am stuck on the factoring. Any help is appreciated.
1 solutions

Answer 150001 by jim_thompson5910(28594)   on 2009-06-10 11:55:24 (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 :
1,2,3,6
-1,-2,-3,-6

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

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

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

First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=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 .

From here, you can use the factored form to solve for "m"

 Rational-functions/199580: b^2-9/b^3-27 i know i have to use the cube root but i stilll dont get how this problem is sloved.. can you please explain this (:1 solutions Answer 149988 by jim_thompson5910(28594)   on 2009-06-10 01:14:42 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get (use the difference of squares formula) Factor to get (use the difference of cubes formula) Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where
 Graphs/199370: What is the equation of the line that contains (7, -9) and (7, 3)?1 solutions Answer 149987 by jim_thompson5910(28594)   on 2009-06-10 00:41:45 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Note: is the first point and is the second point . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Remember, you cannot divide by zero. So this means that the slope is undefined. Since the slope is undefined, this means that the equation of the line through the points and is (which is a vertical line).
 Subset/199545: I review the one that was done with vertice. This one is alittle different please help. Create a graph with five vertices, no bridge and a loop. Thank you so much Terri 1 solutions Answer 149985 by jim_thompson5910(28594)   on 2009-06-10 00:38:32 (Show Source): You can put this solution on YOUR website!Start with five randomly drawn points. These are the vertices Now simply connect every vertex with a line. Make sure that EVERY point has two lines connecting to it (so a bridge doesn't form). In addition to connecting every vertex, draw a line from any one vertex to itself to form a loop (in green):
 Geometric_formulas/199477: Create a graph with five vertices, no bridge, and a loop. 1 solutions Answer 149984 by jim_thompson5910(28594)   on 2009-06-10 00:37:02 (Show Source): You can put this solution on YOUR website!Start with five randomly drawn points. These are the vertices Now simply connect every vertex with a line. Make sure that EVERY point has two lines connecting to it (so a bridge doesn't form). In addition to connecting every vertex, draw a line from any one vertex to itself to form a loop (in green):
 Trigonometry-basics/199577: prove this identity (5 marks): sin(x) + sin(x)cot(x)^2 = csc(x) thanks in advance!1 solutions Answer 149980 by jim_thompson5910(28594)   on 2009-06-10 00:19:38 (Show Source): You can put this solution on YOUR website! ... Start with the given equation. ... Factor out the GCF ... Replace the terms in the parenthesis with . Note: ... Use the identity ... Multiply ... Factor ... Cancel out the common terms. ... Use the identity So we've proven the identity.
 Linear-systems/199578: Solve the sytem of equations by graphing. Then classify the system. x+y=15 x-y=31 solutions Answer 149979 by jim_thompson5910(28594)   on 2009-06-10 00:13:42 (Show Source): You can put this solution on YOUR website!Start with the given system of equations: In order to graph these equations, we must solve for y first. Let's graph the first equation: Start with the first equation. Subtract from both sides. Rearrange the terms and simplify. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is -1 and the run is 1. This means that to go from point to point, we can go down 1 and over 1 So starting at , go down 1 unit and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and ------------------------------------------------------------------------------ Now let's graph the second equation: Start with the second equation. Subtract from both sides. Divide both sides by to isolate . Rearrange the terms and simplify. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 1 and the run is 1. This means that to go from point to point, we can go up 1 and over 1 So starting at , go up 1 unit and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and ------------------------------------------------------------------------------ Now let's graph the two equations together: Graph of (red). Graph of (green) From the graph, we can see that the two lines intersect at the point . So the solution to the system of equations is . This tells us that the system of equations is consistent and independent.
 Money_Word_Problems/199561: 2. A business invests $8,000 in a savings account for two years. At the beginning of the second year, an additional$2,500 is invested. At the end of the second year, the account balance is $11,445. What was the annual interest rate? Im not good at interest 1 solutions Answer 149978 by jim_thompson5910(28594) on 2009-06-10 00:12:23 (Show Source): You can put this solution on YOUR website! Start with the interest formula. Note: since we're only concerned with year intervals, this means that (so we don't have to worry about exponents) Plug in (since$8,000 is the original amount in the account) Distribute So after one full year, you now have dollars in the account. Now because "an additional $2,500 is invested", this pushes up the amount to (just add 2500 to the last expression). Combine like terms to get So the new principal is Since after two years you have$11,445, this means that Go back to the original formula Plug in , and FOIL Subtract 11445 from both sides. Combine like terms. Notice we have a quadratic in the form of where , , and Let's use the quadratic formula to solve for r Start with the quadratic formula Plug in , , and Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Divide So the possible answers are or However, since you CANNOT have a negative interest rate, this means that the only possible answer is Now multiply by 100 to get . ============================================= Answer: So the annual interest rate is 5%
 Proofs/199553: If you could help me with the problem I would really appreciate it. I finished it in 19 lines but i think I did a few steps wrong. 1. B=W 2. L>B 3. ~Bv~W /~(WvL)1 solutions Answer 149975 by jim_thompson5910(28594)   on 2009-06-10 00:06:19 (Show Source): You can put this solution on YOUR website!Note: I'm going to use "<->" for "if and only if" and "*" for "and" 1. B <-> W 2. L -> B 3. ~B v ~W :. ~(W v L) ----------------------------------- 4. (B -> W) * (W -> B) 1 Material Equivalence 5. (W -> B) * (B -> W) 4 Commutation 6. B -> W 4 Simplification 7. W -> B 5 Simplification 8. B -> ~W 3 Material Implication 9. W -> ~W 7,8 Hypothetical Syllogism 10. ~W v ~W 9 Material Implication 11. ~W 10 Tautology 12. ~B 6,11 Modus Tollens 13. ~L 2,12 Modus Tollens 14. ~W * ~L 11,13 Conjunction 15. ~(W v L) 14 De Morgan's Law