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 Expressions-with-variables/81347: Solve I used LCD and got (1x/(x^2 +2x))- (2x +4)/(x^2 + 2x) =3 Then multiplied bith sides by x^2 -2x 3x^2 - 5x -41 solutions Answer 58288 by bucky(2189)   on 2007-05-09 09:15:20 (Show Source): You can put this solution on YOUR website!Given: . . I used LCD and got: . <=== . Then multiplied both sides by <=== should be + between terms, not - . You should now have: . . Multiply out the right side and you get: . . Combine the two terms on the left side that both contain x to get: . . Add x + 4 to both sides to eliminate the terms on the left side. The result is: . . Combine the two terms on the right side that contain x: . . Transpose (switch sides): . . This equation is now in standard quadratic form. It can be solved by graphing, by factoring, or by using the quadratic formula. The more general approach is to use the quadratic formula, but in this case factoring works. The left side of this equation factors to: . . Notice that this equation will be true if either of the factors on the left side equals zero because zero times anything is zero. So, one at a time, set the two factors equal to zero and solve for x. . . Subtract 4 from both sides: . . Divide by 3: . . That's one solution for x. Now set the other factor equal to zero: . . Subtract 1 from both sides: . . That's the second solution for x. . In summary, the two solutions for x are and . You can check both of these by substituting them (one at a time) for x in the original given equation and making sure that the equation still balances on both sides. . Hope this helps you to understand the problem and also to correct your minor mistake. . Cheers ...
 Quadratic_Equations/81000: can i please have help to solve this? (x-1)^2=51 solutions Answer 58101 by bucky(2189)   on 2007-05-07 00:50:58 (Show Source): You can put this solution on YOUR website!Given: . . Square the left side and the equation becomes: . . Then get this equation into standard quadratic form by eliminating the 5 on the right side so that the right side becomes 0. Do this by subtracting 5 from both sides. When you do that subtraction the equation becomes: . . The left side of this equation does not factor nicely. So use the quadratic formula. This formula says that for a quadratic equation of the standard form: . . The values of x that satisfy this equation are given by the equation: . . By comparing our equation with the standard form you can see that a = 1, b = -2, and c = -4. . By substituting these values into the equation that defines the values of x you get: . . This simplifies to: . . Note that . Substituting for reduces the equation for x to: . . So this problem has two solutions for x ... and . Hope this helps you to understand the problem a little better. .
 Quadratic_Equations/80954: Write the equation of the line with slope 2 and y-intercept (0, 0). Then graph the line.1 solutions Answer 58081 by bucky(2189)   on 2007-05-06 19:47:39 (Show Source): You can put this solution on YOUR website!For this problem you can use the slope-intercept equation. This equation has the form: . . In this equation m represents the slope and b is the value on the y-axis where the graph intersects the y-axis. . In this problem you are told that the slope is -2. And you are told that the graph goes through the origin. This means that the graph intersects the y-axis at a y value of 0. This translates to m = -2 and b = 0. Substituting these values into the slope-intercept equation results in the equation becoming" . . You can then simplify it by dropping the 0 to get the equation: . . The fact that the slope is negative tells you that the graph goes down as you move to the right. . You can now get some points on the graph by assigning values to x and computing the corresponding values of y. For example, let x = -5. Plug that value in for x and the equation becomes: . . So when x is -5, then y = +10. This means the point (-5, +10) is on the graph. . The problem tells you that (0, 0) is on the graph. . Then suppose that we let x = +5. Substituting this value into the equation results in: . . This means that when x equals +5, the corresponding value of y is -10. So the point (+5, -10) is on the graph. If you plot these three points you should see that they lie on a straight line. Take a straight edge and line it on these points. Then extend a line through all three points. When you do you should have a graph that looks like: . . Hope this helps you to understand the problem and see how to get an answer. . Cheers .
 Graphs/80938: The problem I have--not from a text book--is this: "Draw the graph of the linear function n(x) = 0" On this one, I am totally lost. Other than understanding there is a "0" point on the y axis, I don't know what I am supposed to do. Help!1 solutions Answer 58080 by bucky(2189)   on 2007-05-06 19:21:09 (Show Source): You can put this solution on YOUR website!It helps to visualize this one if you replace n(x) with y to get: . y = 0 . Now think of it this way: since there is no x in the function, y is always zero regardless of what value you assign to x. Is (-10, 0) a solution to this equation? Yes it is because y equals zero. So is (0,0) a solution set because although x is zero, the critical part is that y = 0 and that satisfies the equation. How about the point (50, 0)? Same thing. . If you plot all of these points, you begin to see that the graph of y = 0 (or its equivalent n(x) = 0) is the x-axis, because the x-axis is the line of all points having a y value of zero. . Similarly, y = -3 is a horizontal line intersecting the y-axis at -3. No matter what value you assign to x, the corresponding value of y must always be -3. . Hope this is one way of looking at the problem that will help you to make some sense out of it.
 Inequalities/80895: This question is from textbook Glenco Mathematics Algebra 1 I am really stumped as to how to solve this inequality. I would appreciate ANY answers. I just, for some reason, cannot work this one. 3a + 8/2 < 10 Thanks so much. :) 1 solutions Answer 58041 by bucky(2189)   on 2007-05-06 12:19:03 (Show Source): You can put this solution on YOUR website!You can work problems of this type just as you would solve an equation with the exception that if you divide or multiply by a negative quantity, then you must reverse the direction of the inequality sign. . So let's start with the given expression: . . Notice that so we can replace it by 4 to get: . . Get rid of the 4 on the left side by subtracting 4 from both sides of the inequality to get: . . Now reduce the left side to just "+a" by dividing both sides of the inequality by +3 to get: . . which simplifies to . The original inequality should be satisfied as long as the value of "a" is less than 2. . Let's build our self confidence by trying some values for "a". Suppose we let "a" equal zero. That value is obviously less than +2. If we substitute zero for a in the original inequality we get: . which becomes . That works. Similarly, if we let a = +1 the original inequality becomes: . and this further simplifies to . That works also. . Now let's set a = +3. That is outside the limit we found since +3 is not less than +2. . With a = +3 the original inequality becomes: . . This simplifies to , and this obviously is not true. . From these spot checks, it seems as though our answer is correct. . Hope this helps you to understand the problem. . Cheers.
 Problems-with-consecutive-odd-even-integers/80893: the sum of a number and four times its reciprocal is -5. what is the number?1 solutions Answer 58037 by bucky(2189)   on 2007-05-06 12:01:40 (Show Source): You can put this solution on YOUR website!Given: . The sum of a number and four times its reciprocal is -5 . The "number" is the unknown, so let's represent it by x. . The reciprocal of the number by definition is 1 divided by the number, and 4 times the reciprocal is, therefore: . . So the sum of the number and 4 times its reciprocal is: . . and the problem tells you that this is -5. So set it equal to -5: . . Now let's solve for x. We can do so by multiplying every term (both sides) of this equation by x to eliminate the denominator. Do that and the equation becomes: . . Add 5x to both sides to eliminate the -5x on the right side and get the equation into the standard quadratic form of: . . There are several ways that this can be solved (graphing; completing the square or its equivalent, using the quadratic formula; but in this case factoring is probably the easiest.) The equation factors to: . . This equation will be true if either of the two factors is equal to zero. So set each factor equal to zero and solve for the value of x that will make that happen: . . Subtract 1 from both sides of the equation and the result is: . . Then set the second factor equal to zero: . . and subtract 4 from both sides to get: . . So there are two possible values for x that will work ... -1 and -4 . Check them out by evaluating each in the original problem. . If x = -1, will ? Substitute -1 for x and you get: . . This value of -1 works. Now let's try the second value, x = -4. Substitute for x and you get: . . This also works. Therefore, your problem has two solutions ... x = -1 and x = -4. . Hope this helps you to understand the problem and how you can work it to a solution. . Cheers
 Quadratic_Equations/80874: solve the equation (3w+4)(2w-7)=0 1 solutions Answer 58034 by bucky(2189)   on 2007-05-06 11:38:18 (Show Source): You can put this solution on YOUR website!Given the equation: . (3w+4)(2w-7)=0 . Solve for w. . Notice that if at least one of the two factors on the left side of the equation is zero, then the equation will be true because zero times anything is zero. . Therefore, the equation will be true if either 3w + 4 = 0 or if 2w - 7 = 0 because that will make the left side equal the right side. . Let's first say that (3w + 4) equals zero. . 3w + 4 = 0 . Solve for w by first subtracting 4 from both sides to get: . 3w = -4 . Then divide both sides by 3 (which is the multiplier of w) to get: . w = -4/3 . So that's one value for w that will make the equation true. . Next let's look at the second factor ... (2w - 7) and set it equal to zero: . 2w - 7 = 0 . Get rid of the -7 on the left side by adding 7 to both sides to get: . 2w = 7 . Solve for w by dividing both sides by 2 and the result is: . w = 7/2 . That's the second value for w that will make the equation true. . In summary, the values of w that will make the equation true are: . w = -4/3 and w = 7/2 . Hope this helps you to understand why setting each of the factors equal to zero will give you two values for w that make the equation work.
 Quadratic_Equations/80883: factor the polynomial completely 121rSQUARED-64tSQUARED1 solutions Answer 58028 by bucky(2189)   on 2007-05-06 11:09:34 (Show Source): You can put this solution on YOUR website!Given: . . The problem is to factor this expression. . The given expression contains the difference of two perfect squares. As such it falls under the factoring rule: . . By comparing this rule with the given expression you can see that: . . and this leads to . Similarly: . . and taking the square root of both sides of this results in: . . Now return to the factoring rule and substitute for a and b using the above expressions. When you do you get: . . and the right side of this equation is the answer you are looking for. . Hope this helps you become familiar with the factoring rule for the difference between squares. It comes up reasonably often in book problems that you need to be aware of it, and the more you see and use it, the more you will tend to remember it. . Cheers ...
 Geometry_Word_Problems/80793: f(x)= x2 - 7x + 10; find (a) f (0), (b) f (5), and (c) f (-2)1 solutions Answer 57956 by bucky(2189)   on 2007-05-05 00:40:59 (Show Source): You can put this solution on YOUR website!Given: . . find (a) f (0), (b) f (5), and (c) f (-2) . Problem (a) tells you to go to the given equation, replace every x with a zero, and then simplify the result. . Replace every x with zero to get: . . So for (a) you have . Problem (b) ... same thing only replace every x with 5 to get: . . So for (b) you have . . For problem (c) ... same thing again only this time replace every x with -2 to get: . . So for (c) you have . Hope this helps you to understand what the problem was asking you to do and how you should go about doing problems like these. .
 Numbers_Word_Problems/80684: The sun of two numbers is 51. twice the first plus 4 times the second is 128. what are the numbers?1 solutions Answer 57840 by bucky(2189)   on 2007-05-04 00:15:55 (Show Source): You can put this solution on YOUR website!Call the first number F and the second number S . The sum of the first number and the second number equals 51. In equation form this is: . F + S = 51 . Then the problem says that twice the first (2*F) plus four times the second (4*S) equals 128. In equation form this is: . 2F + 4S = 128 . So we now have a set of two equations: . F + S = 51 2F + 4S = 128 . We can solve this set of equations by the process of variable elimination. One way is to multiply the to equation (all terms on both sides) by -2 to get: . -2F - 2S = -102 2F + 4S = 128 . Now add the two equations vertically. When you do, the -2F in the top equation cancels the 2F in the bottom equation. Continuing with the vertical addition you get: . 2S = 26 . Dividing both sides of this equation by 2 results in: . S = 13 . Now return to the original first equation which said that the sum of the first and second numbers is 51 ... . F + S = 51 . Substitute 13 for S and this equation becomes: . F + 13 = 51 . Solve for F by subtracting 13 from both sides: . F = 38 . In summary, the two numbers are 13 and 38 . We already know that 13 plus 38 equals 51 as required by the problem. . Then 2 times the first is 2 * 38 and that equals 76. Add to that 4 times 13 which is 52. The result of 76 + 52 is 128 as was also required by the problem. The answer checks. . Hope this helps you to understand the problem. .
 Travel_Word_Problems/80682: Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle. 1 solutions Answer 57837 by bucky(2189)   on 2007-05-03 23:53:22 (Show Source): You can put this solution on YOUR website!For this equation you use the formula: . D = R*T . where D = the distance traveled, R = the rate or speed, and T = the time . The rate and the time are the two unknowns. For the 200 miles we can write the equation for the actual trip as: . 200 = R * T <--- call this the first equation . For the proposed trip you know that the new rate equals the old rate plus 10 mph. You can write this new rate as (R + 10). . And you also know that the new time is an hour less than the old time. You can write this new time as (T - 1) . Therefore you can write the equation for the new trip as: . 200 = (R + 10)*(T - 1) . Multiply the right side out to get: . 200 = R*T - R + 10*T - 10 <--- call this the second equation . Since you need to solve this second equation for R, solve the first equation for T in terms of R and then substitute that into this second equation. . Solving the first equation for T in terms of R you divide both sides by R and you get: . 200 = T * R . 200/R = (T*R)/R . 200/R = T . Now substitute the left side of this equation into the second equation to get: . 200 = R*(200/R) - R + 10*(200/R) - 10 . Simplify this by multiplying out the right side: . 200 = 200 - R + (2000/R) - 10 . Subtract 200 from both sides and the equation becomes: . 0 = - R + 2000/R - 10 . Multiply both sides by -R and you get: . 0 = R^2 - 2000R/R + 10R . The middle term on the right side simplifies to -2000 and the equation becomes: . 0 = R^2 - 2000 + 10R . Transpose this equation (switch sides) and rearrange terms so it is in the more conventional form of: . R^2 + 10R - 2000 = 0 . This equation factors into: . (R + 50)*(R - 40) = 0 . This equation will be true if either factor on the left side equals zero. So set each equal to zero and solve for R: . R + 50 = 0 . Subtract 50 from both sides and R becomes: . R = -50 mph . Then set the second factor equal to zero: . R - 40 = 0 . Add 40 to both sides to get: . R = 40 mph . Ignore the first answer of -50 mph because a negative speed doesn't really make sense. . So the answer is 40 mph as the speed. . Check this out. At 40 mph you drive the 200 miles in 5 hours. . Now increase the speed by 10 mph to 50 mph. If you drive 200 miles at 50 mph it will take 4 hours. The answer checks ... increasing the speed to 50 mph reduces the time it takes to drive 200 miles by 1 hour. . So the answer to the original rate is 40 mph. . Hope this helps you to understand the problem and a way that you can solve it. .
 Evaluation_Word_Problems/80518: An employee who produces x units per hour earns an hourly wage of y=0.55x+7(in dollars) find the hourly wage for an employee who produces 13 units per hour.1 solutions Answer 57765 by bucky(2189)   on 2007-05-03 03:52:44 (Show Source): You can put this solution on YOUR website!Given: . . in which y equals the hourly wage in dollars and x is the number of units produced per hour. . You are asked to find the hourly wage of a worker who produces 13 units per hour. . All you have to do is to refer to the equation and substitute 13 for x. If you do that, you get: . . Do the multiplication on the right side and the equation becomes: . . Now add the two terms on the right side of the equation to get: . . That's the answer. A worker who produces 13 units per hour earns a wage of \$14.15 per hour. . Hope this helps you to understand the problem a little better. .
 Linear_Equations_And_Systems_Word_Problems/80526: cost of gravel a gravel dealer charges \$50 plus \$30 per cubic yard for delivering a truckload of gravel. Express the total cost C(n) in dollars as a function of the number of cubic yards delivered n. Find C(12). 1 solutions Answer 57764 by bucky(2189)   on 2007-05-03 03:40:21 (Show Source): You can put this solution on YOUR website!Call n the number of cubic yards that are ordered. Then to find the cost of just the gravel you need to multiply the number of cubic yards (n) by \$30 per yard. Therefore, that cost is . To that cost the dealer will add \$50 for delivery. This will make the total cost of gravel plus delivery [call it C(n)]: . dollars. . This is one of the answers you were asked for in the problem. . Using this equation you can calculate the the cost of 12 delivered cubic yards by replacing n with 12. This results in: . . Multiply 30 by 12 and the equation becomes: . . So 12 cubic yards of delivered gravel costs \$410. . Hope this helps you to understand the problem a little more. .
 logarithm/80458: This question is from textbook Solve the equation: 1n x = 1n 2 + 1n (3x - 1) I know the answer is 2/5 from the answers in the back of the book. I need to know how the solution was solved. I really appreciate your help. 1 solutions Answer 57763 by bucky(2189)   on 2007-05-03 03:16:15 (Show Source): You can put this solution on YOUR website!Given: . 1n = 1n + 1n . Collect the logarithms that contain x on one side of the equation. You can do that by subtracting ln from both sides of the equation. When you do that subtraction you get: . ln - ln = ln . Now apply the rule that the difference in two logarithms is the same as the log of their division. In equation form that rule says: . ln - ln = ln . With that rule the equation becomes: . ln = ln . Look at this carefully ... comparing the left side to the right side. For this to be true the two logarithms must be equal, and this means that: . . To get rid of the denominator, multiply both sides by the denominator of to get: . . On the left side the multiplier cancels with the denominator. And multiplying the right side, gives: . . and it reduces to: . . Solve this as you would any equation. You need to gather all the terms that contain x on one side of the equation and everything else on the other side. You can do this by subtracting 6x from both sides to get: . . combine the terms on the left side and the result is: . . solve for x by dividing both sides by -5 and the outcome is: . . and that's how to get the book answer. Hope this helps to familiarize you with logarithms.