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 Coordinate-system/139504: Find the domain of the function f(x)=1 _____ x-41 solutions Answer 101723 by jim_thompson5910(28536)   on 2008-05-01 11:40:42 (Show Source): You can put this solution on YOUR website! Start with the given function Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain. Add 4 to both sides Combine like terms on the right side Since makes the denominator equal to zero, this means we must exclude from our domain So our domain is: which in plain English reads: x is the set of all real numbers except So our domain looks like this in interval notation note: remember, the parenthesis excludes 4 from the domain If we wanted to graph the domain on a number line, we would get: Graph of the domain in blue and the excluded value represented by open circle Notice we have a continuous line until we get to the hole at (which is represented by the open circle). This graphically represents our domain in which x can be any number except x cannot equal 4
 Equations/139536: -5(x-2)+3=51 solutions Answer 101722 by jim_thompson5910(28536)   on 2008-05-01 11:39:40 (Show Source): You can put this solution on YOUR website! Start with the given equation Distribute Combine like terms on the left side Subtract 13 from both sides Combine like terms on the right side Divide both sides by -5 to isolate x Reduce -------------------------------------------------------------- Answer: So our answer is (which is approximately in decimal form)
 Coordinate-system/139503: Graph 3x-2y=-181 solutions Answer 101721 by jim_thompson5910(28536)   on 2008-05-01 11:38:59 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce 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 2. This means that to go from point to point, we can go up 3 and over 2 So starting at , go up 3 units and to the right 2 units 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
Quadratic_Equations/139519: What do I solve this by factoring?
20x^2 + 13x + 2
Thanks!
1 solutions

Answer 101720 by jim_thompson5910(28536)   on 2008-05-01 11:34:57 (Show Source):
You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 20 and 2 respectively.

Now multiply the first coefficient 20 and the last coefficient 2 to get 40. Now what two numbers multiply to 40 and add to the middle coefficient 13? Let's list all of the factors of 40:

Factors of 40:
1,2,4,5,8,10,20,40

-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 40
1*40
2*20
4*10
5*8
(-1)*(-40)
(-2)*(-20)
(-4)*(-10)
(-5)*(-8)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 13

First NumberSecond NumberSum
1401+40=41
2202+20=22
4104+10=14
585+8=13
-1-40-1+(-40)=-41
-2-20-2+(-20)=-22
-4-10-4+(-10)=-14
-5-8-5+(-8)=-13

From this list we can see that 5 and 8 add up to 13 and multiply to 40

Now looking at the expression , replace with (notice adds up 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 )

Set the factorization equal to zero

Now set each factor equal to zero:
or

or Now solve for x in each case

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

So our solutions are

or

Quadratic_Equations/139520: What do I solve this by factoring?
49x^2 - 14x - 3
Thanks!
1 solutions

Answer 101719 by jim_thompson5910(28536)   on 2008-05-01 11:33:05 (Show Source):
You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 49 and -3 respectively.

Now multiply the first coefficient 49 and the last coefficient -3 to get -147. Now what two numbers multiply to -147 and add to the middle coefficient -14? Let's list all of the factors of -147:

Factors of -147:
1,3,7,21,49,147

-1,-3,-7,-21,-49,-147 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -147
(1)*(-147)
(3)*(-49)
(7)*(-21)
(-1)*(147)
(-3)*(49)
(-7)*(21)

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

Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14

First NumberSecond NumberSum
1-1471+(-147)=-146
3-493+(-49)=-46
7-217+(-21)=-14
-1147-1+147=146
-349-3+49=46
-721-7+21=14

From this list we can see that 7 and -21 add up to -14 and multiply to -147

Now looking at the expression , replace with (notice adds up 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 )

Set the factorization equal to zero

Now set each factor equal to zero:
or

or Now solve for x in each case

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

So our solutions are

or

 Numeric_Fractions/139523: 4/5 = -11r - 2/31 solutions Answer 101718 by jim_thompson5910(28536)   on 2008-05-01 11:29:50 (Show Source): You can put this solution on YOUR website! Start with the given equation Multiply both sides by the LCM of 15. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver) Distribute and multiply the LCM to each side Subtract 12 from both sides Add 165r to both sides Combine like terms on the right side Divide both sides by 165 to isolate r Reduce -------------------------------------------------------------- Answer: So our answer is (which is approximately in decimal form)
 Graphs/139533: Graph the line with slope 1/2 passing through the point (-3,3) . 1 solutions Answer 101717 by jim_thompson5910(28536)   on 2008-05-01 11:28:24 (Show Source): You can put this solution on YOUR website! If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Distribute Multiply and to get Add 3 to both sides to isolate y Combine like terms and to get (note: if you need help with combining fractions, check out this solver) ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (,) is: which is now in form where the slope is and the y-intercept is Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver) Graph of through the point (,) and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.
 Functions/138844: k. What price would you sell the tile sets at to realize this profit (hint, use the demand equation from part a)? 1 solutions Answer 101293 by jim_thompson5910(28536)   on 2008-04-27 16:46:55 (Show Source): You can put this solution on YOUR website!From part a), we found the demand equation where x is the number of units sold and p is the price. To find the demand simply plug in (remember, we max out the profit when 28 tiles are sold) Start with the given equation Plug in Add So the price should be set at $34  Trigonometry-basics/138826: Write as a sum Write as a product1 solutions Answer 101283 by jim_thompson5910(28536) on 2008-04-27 14:46:01 (Show Source): You can put this solution on YOUR website!1) Start with the given expression Use the Product-to-Sum formula Add/Subtract Distribute So 2) Are you sure that it's ? There are no corresponding sum-to-product identities for mixed the trig functions sine and cosine. Double check your assignment.  Trigonometry-basics/138824: Solve 1 solutions Answer 101281 by jim_thompson5910(28536) on 2008-04-27 14:38:55 (Show Source): You can put this solution on YOUR website! Start with the given equation Let . So this means that Replace with u and with 2u Rewrite as Replace with Square Factor out Now use the zero product property: ...or... Now let's solve : Take the square root of both sides ...or... Take the arcsine of both sides Since , this means ...or... However, since is not in the interval [0,2), it is not a solution. So the first solution is -------------------------------------------------------------------- Now let's solve : Subtract 1 from both sides Divide both sides by -4 Take the square root of both sides: ...or... So let's solve the first part : ...or... ...or... However since our interval is positive, the negative answer is not in the interval. So another part of the solution is --------------------- Now let's solve the second part : ...or... ...or... So another part of the solution is =============================================== Answer: So all together our solutions are:  Trigonometry-basics/138823: Find the exact value of 1 solutions Answer 101279 by jim_thompson5910(28536) on 2008-04-27 14:35:46 (Show Source): You can put this solution on YOUR website!let so Substitute "2u" in for "u" Multiply Take the arccosine of the cosine of to get Replace with Add 1 to both sides Divide both sides by 2 Take the square root of both sides Simplify Rationalize the denominator if necessary So  Trigonometry-basics/138821: Prove the identity: (x is theta)1 solutions Answer 101276 by jim_thompson5910(28536) on 2008-04-27 14:25:07 (Show Source): You can put this solution on YOUR website!Note: I'm only manipulating the left side. I'm not touching the right side. I'm only showing it for comparison. Start with the given equation Break up to get Expand using the sum-difference formulas Replace with . Replace with . Factor out Rearrange the terms Factor out 2 Replace with Replace with Distribute Combine like terms Distribute So we've just proven that is an identity.  Trigonometry-basics/138819: Prove the identity: (x is theta)1 solutions Answer 101275 by jim_thompson5910(28536) on 2008-04-27 14:16:58 (Show Source): You can put this solution on YOUR website!Note: I'm only manipulating the left side. I'm not touching the right side. I'm only showing it for comparison. Start with the given equation Expand by using the Sum-Difference Formulas Foil Combine like terms Replace with . Replace with Distribute Combine like terms So we've just proven that is an identity.  Trigonometry-basics/138815: Prove the identity: (x is theta)1 solutions Answer 101273 by jim_thompson5910(28536) on 2008-04-27 14:08:28 (Show Source): You can put this solution on YOUR website!Note: I'm only manipulating the left side. I'm not touching the right side. I'm only showing it for comparison. Start with the given equation Replace with Simplify Multiply both the numerator and denominator by the LCD . This will clear out the denominators in both the numerator and the denominator. Distribute and multiply Replace with 1 Simplify So we've just proven that is an identity.  Trigonometry-basics/138801: Graph and the period and phase shift1 solutions Answer 101260 by jim_thompson5910(28536) on 2008-04-27 12:41:30 (Show Source): You can put this solution on YOUR website!Graph of (not the best graph, but it gives you an idea of what it looks like) Remember the general form is where the period is , and the phase shift is So in this case... the period is , the phase shift is  Trigonometry-basics/138798: Graph and find amplitude, period and phase shift1 solutions Answer 101259 by jim_thompson5910(28536) on 2008-04-27 12:36:21 (Show Source): You can put this solution on YOUR website!Graph of Remember the general form is where the amplitude is , the period is , and the phase shift is So in this case... the amplitude is , the period is , the phase shift is  Functions/138710: "A company calculates it's profit by finding the difference between revenue and cost. The cost function of producing x hammers is C(x) = 4x+170. If each hammer is sold for$10, the revenue function for selling x hammers is R (x) = 10x." 1.) How many hammers must be sold to make a profit? 2.) How many hammers must be sold to make a profit of $100? 1 solutions Answer 101212 by jim_thompson5910(28536) on 2008-04-26 17:58:25 (Show Source): You can put this solution on YOUR website!1) The profit function is defined by: Profit = Revenue - Cost So in function notation it looks like: So in our case: Plug in and Distribute the negative Combine like terms Set the right side greater than zero. Remember we're looking for positive profit. Add 170 to both sides Divide both sides by 6 Divide Round to the nearest whole number. A third of a hammer can't be sold. So when , we'll have positive profit. So more than 29 hammers must be sold to gain a profit. 2) We're still using the profit function. So Since we want a profit of$100, simply plug in . Start with the profit function Plug in Add 170 to both sides Divide both sides by 6 So when , we'll have a profit of $100. So 45 must be sold to make a profit of$100.
 Graphs/138688: The demand equation is given as and the supply as . Find the equilibrium price.1 solutions Answer 101205 by jim_thompson5910(28536)   on 2008-04-26 14:31:24 (Show Source): You can put this solution on YOUR website! Start with the given system Now move onto the second equation Plug in Multiply both sides by the LCD 50x Distribute and multiply Subtract 625000 from both sides. Factor the left side (note: if you need help with factoring, check out this solver) Now set each factor equal to zero: or or Now solve for x in each case So our answer is or However, a negative number of jackets does not make sense. So the only solution is Now plug into the second equation So our solution is and So when 500 jackets are sold, the supply will equal the demand at $25.  Graphs/138687: Graph the two equations and and find the solutions1 solutions Answer 101203 by jim_thompson5910(28536) on 2008-04-26 14:21:22 (Show Source): You can put this solution on YOUR website! Start with the given equation Solve for y or Break up the right side Now let's graph Graph of Now let's graph Graph of Now graph Graph of Now graph the three equations together (note: the first two equations make up ) Graph of (red). Graph of (green). Graph of (blue) From the graph, we can see that there are no intersections. So there are no solutions.  Graphs/138686: Graph the two equations and and find the solutions1 solutions Answer 101201 by jim_thompson5910(28536) on 2008-04-26 14:12:50 (Show Source): You can put this solution on YOUR website!First graph Graph of Now graph Graph of Now graph the two equations together Graph of (red. Graph of (green) From the graph, we can see that the two equations intersect at the points (-2,1) and (1,4) So this means that the solutions are and OR and Graphs/138685: A cost function is given as . Find the cost of producing 100 boxes 1 solutions Answer 101199 by jim_thompson5910(28536) on 2008-04-26 14:00:53 (Show Source): You can put this solution on YOUR website! Since we're trying to find , our test zero is 100 Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.  100 | 1/2 -8 -80 200 | Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)  100 | 1/2 -8 -80 200 | 1/2 Multiply 100 by 1/2 and place the product (which is 50) right underneath the second coefficient (which is -8)  100 | 1/2 -8 -80 200 | 50 1/2 Add 50 and -8 to get 42. Place the sum right underneath 50.  100 | 1/2 -8 -80 200 | 50 1/2 42 Multiply 100 by 42 and place the product (which is 4200) right underneath the third coefficient (which is -80)  100 | 1/2 -8 -80 200 | 50 4200 1/2 42 4120 Add 4200 and -80 to get 4120. Place the sum right underneath 4200.  100 | 1/2 -8 -80 200 | 50 4200 1/2 42 4120 Multiply 100 by 4120 and place the product (which is 412000) right underneath the fourth coefficient (which is 200)  100 | 1/2 -8 -80 200 | 50 4200 412000 1 42 4120 Add 412000 and 200 to get 412200. Place the sum right underneath 412000.  100 | 1/2 -8 -80 200 | 50 4200 412000 1/2 42 4120 412200 Since the last column adds to 412200, we have a remainder of 412200. ------------------------------------ Answer: So according to the remainder theorem, So the cost to produce 100 boxes is$412,200