Solve the following equation. Show all your work
x/x-2+x-1/x+1=-1
Do you first do (x-2)(x+1)? I need help. I can't seem to get the idea of what to do.
Thanks
Yes, just as when you add 2/3 and 5/7, you need to find the common denominator. In this case, that is indeed (x-2)(x+1)
So, multiplying by (x-2)(x+1), the common factors cancel, and you are left with
x(x+1) + (x-1)(x+2) = -1(x-2)(x+1)
You can now ignore the denominator, because all the terms have the same denominator. The fractions add up if the new numerators add up.
The only thing you have to watch out for is extraneous roots. Because the original fractions had (x-2) and (x+1) in the denominator, those fractions are undefined if x=2 or x = -1. So if one of your solutions is -1 or 2, it must be discarded.
Thank you for helping!
To solve the equation x/x-2 + x-1/x+1 = -1, we need to find a common denominator for the fractions on the left side of the equation.
Step 1: Find the common denominator
The denominators of the two fractions are (x-2) and (x+1). To find the common denominator, we multiply the denominators together: (x-2)(x+1).
Step 2: Multiply each fraction by the missing factor to get the common denominator
To get the common denominator (x-2)(x+1), we need to multiply the first fraction x/x-2 by (x+1)/(x+1) and the second fraction x-1/x+1 by (x-2)/(x-2). This gives us:
(x(x+1))/((x-2)(x+1)) + ((x-1)(x-2))/((x-2)(x+1)) = -1
Step 3: Simplify the fractions on the left side
Expanding the numerators, we have:
(x^2 + x)/((x-2)(x+1)) + (x^2 - 3x + 2)/((x-2)(x+1)) = -1
Step 4: Combine the fractions on the left side
Since the denominators are the same, we can add the numerators together:
(x^2 + x + x^2 - 3x + 2)/((x-2)(x+1)) = -1
Simplifying the numerator:
(2x^2 - 2x + 2)/((x-2)(x+1)) = -1
Step 5: Multiply both sides of the equation by the common denominator
To eliminate the denominator, we multiply both sides of the equation by (x-2)(x+1):
(2x^2 - 2x + 2) = -1 * (x-2)(x+1)
Step 6: Expand and simplify the equation
Expanding the right side of the equation:
2x^2 - 2x + 2 = -x^2 + 3x - 2
Step 7: Move all terms to one side of the equation
Rearranging the terms:
3x^2 - 5x + 4 = 0
This is a quadratic equation. You could then apply the quadratic formula or factoring to find the solutions for x.