What should you do If you encounter an equation that hat a fraction in it? In th

Question

What should you do If you encounter an equation that hat a fraction in it? In this section, we are going to expand on that idea to involve equations that have variables in the denominator. These are called rational equations. Take the equation 3/x + 5 = 1/4. Taking into account all of the denominators in the equation, what is their LCD? Write the LCD next to each of the numerators, cancel the denominators and solve. Be sure to use parentheses when necessary. Now take the equation 5/x^2 - 4 = 7/x - 2 Looking at the first denominator, what can you do to it? What is the LCD for this equation? Write the LCD next to each of the numerators, cancel the denominators and solve. Be sure to use parentheses when necessary. So, whenever solving a rational equation, you must factor each denominator first. Imagine that you have an equation whose denominators are (x + 1)^2 and (x + 1)(x - 1) Would the LCD be (x + 1)(x - 1)? Explain your answer.

Explanation / Answer

If we encounter an equation, that has a fraction in it, we should multiply both the sides of the equation by the denominator of the fraction to simplify the equation. The denominators are x and 4. Their LCD is 4x. 3/x +5= ¼ or, (3+5x)/x = ¼ or, (3+5x)*4x/x = 4x/4 or, 4(3+5x) = x or, 12+20x = x or, 20x-x = -12 or, 19x = -12 so that x = -12/19. 5/(x2-4)= 7/(x-2). Here, x2-4 = (x+2)(x-2). Thus, we factorize x2-4 before computing the LCD. The LCD is (x+2)(x-2) = x2-4. We have 5/(x2-4)= 7/(x-2) or, [5 (x2 -4)/(x2-4)= [7(x2 -4)/(x-2)] or, 5 = [7(x+2)(x-2)]/(x-2) or, 5 = 7(x+2) or, 5 = 7x+14 or, 7x = -14+5 = -9. Therefore, x = -9/7. The LCD of (x+1)2 and (x+1)(x-1) is (x+1)2(x-1). as (x+1)(x-1) is not divisible by(x+1)2 .

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