Rational EquationsDate: 06/07/97 at 23:06:53 From: Catherine McDonald Subject: Polynomials I'm in 9th grade and I have a final coming up. I thought I'd try to understand how to solve rational equations before I get to the final. For something like (1/x+1) - (1/x+2) = 1/2, I understand the cross multiplying part, but then what do you do? One teacher said that you had to find the common multiple, but then another teacher said something different. Please try to explain this problem. Another example would be fantastic. Thanks in advance, Catherine Date: 06/08/97 at 10:52:22 From: Doctor Anthony Subject: Re: Polynomials Dear Catherine, Be careful how you write down the fractions because the method you use could be interpreted as 1/x + 1 - 1/x - 2, which I am sure you did not intend. We have: 1/(x+1) - 1/(x+2) = 1/2 To add fractions you must have the same denominator in each fraction. An example from arithmetic would be: 1/2 + 1/3. The common denominator will have to be a number into which both 2 and 3 will divide. Obviously 6 is such a number. To make the denominator 6 in the fraction 1/2, multiply top and bottom by 3 to get 3/6. To make the denominator 6 in the fraction 1/3, multiply top and bottom by 2 to get 2/6. We now have to add 3/6 + 2/6, and since the denominators are the same we can add the numerators to get: (3 + 2)/6 = 5/6 On paper you would write this: 3 + 2 5 --------- = ----- 6 6 Looking at the algebra problem, the common denominator will be the product (x+1)(x+2). Since we multiply the denominator of the first fraction by (x+2), we must multiply the numerator also by (x+2). The first fraction could be written: x+2 ---------- (x+1)(x+2) The second fraction could be written: x+1 ---------- (x+1)(x+2) In practice you would write this: (x+2) - (x+1) 1 -------------- = ----------- (x+1)(x+2) (x+1)(x+2) Now we can start to solve the equation: 1 --------- = 1/2 (x+1)(x+2) Cross multiplying: 2 = (x+1)(x+2) 2 = x^2 + 3x + 2 0 = x^2 + 3x 0 = x(x+3) So we have two possible values of x which satisfy the equation; either x = 0 or x = -3. We can check if these are correct: If x = 0, the equation becomes 1/1 - 1/2 = 1/2, which is correct. If x = -3, the equation becomes 1/-2 - 1/-1 = -1/2 + 1 = 1/2, again correct. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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