Question
Show all you work.
x over x-2 + x-1 over x+1 = -1
x over x-2 + x-1 over x+1 = -1
Answers
GPT 3.5
To solve this equation, we first need to find a common denominator for the two fractions on the left side of the equation. The common denominator for x - 2 and x + 1 is (x - 2)(x + 1).
Therefore, the equation becomes:
x(x + 1) / [(x - 2)(x + 1)] + (x - 1)(x - 2) / [(x - 2)(x + 1)] = -1
Now, we combine the fractions:
[x(x + 1) + (x - 1)(x - 2)] / [(x - 2)(x + 1)] = -1
Now, expand the numerator and simplify the equation further:
[x^2 + x + x^2 - 3x + 2] / [(x - 2)(x + 1)] = -1
[2x^2 - 2x + 2] / [(x - 2)(x + 1)] = -1
Now, multiply both sides of the equation by [(x - 2)(x + 1)] to eliminate the denominator:
2x^2 - 2x + 2 = -1(x - 2)(x + 1)
2x^2 - 2x + 2 = -x^2 - x + 2
Now, combine like terms and simplify:
2x^2 - 2x + 2 = -x^2 - x + 2
2x^2 - 2x + 2 = -x^2 - x + 2
Now, add x^2 and x to both sides of the equation:
2x^2 - 2x + x^2 + x = 2
3x^2 - x = 2
Finally, rearrange the equation to get it in standard form:
3x^2 - x - 2 = 0
This is the quadratic equation that needs to be solved to find the value(s) of x that satisfy the original equation. You can solve it using the quadratic formula or factoring techniques.
Therefore, the equation becomes:
x(x + 1) / [(x - 2)(x + 1)] + (x - 1)(x - 2) / [(x - 2)(x + 1)] = -1
Now, we combine the fractions:
[x(x + 1) + (x - 1)(x - 2)] / [(x - 2)(x + 1)] = -1
Now, expand the numerator and simplify the equation further:
[x^2 + x + x^2 - 3x + 2] / [(x - 2)(x + 1)] = -1
[2x^2 - 2x + 2] / [(x - 2)(x + 1)] = -1
Now, multiply both sides of the equation by [(x - 2)(x + 1)] to eliminate the denominator:
2x^2 - 2x + 2 = -1(x - 2)(x + 1)
2x^2 - 2x + 2 = -x^2 - x + 2
Now, combine like terms and simplify:
2x^2 - 2x + 2 = -x^2 - x + 2
2x^2 - 2x + 2 = -x^2 - x + 2
Now, add x^2 and x to both sides of the equation:
2x^2 - 2x + x^2 + x = 2
3x^2 - x = 2
Finally, rearrange the equation to get it in standard form:
3x^2 - x - 2 = 0
This is the quadratic equation that needs to be solved to find the value(s) of x that satisfy the original equation. You can solve it using the quadratic formula or factoring techniques.
You need to use the foil method (first outside, then inside last.)
Multiply the first two terms together (x*x) then the outside terms (x*1,) inside (-2*)
Simplify and combine like terms, the continue to solve the equation and find the solution for X.
Multiply the first two terms together (x*x) then the outside terms (x*1,) inside (-2*)
Simplify and combine like terms, the continue to solve the equation and find the solution for X.
GPT 3.5
I apologize for the oversight. Let's redo the previous steps using the FOIL method:
Given equation: (x / (x - 2)) + ((x - 1) / (x + 1)) = -1
Multiplying the first two terms: x * x = x^2
Multiplying the outside terms: x * 1 = x
Multiplying the inside terms: -2 * x = -2x
The equation now becomes: (x^2 + x - 2x) / (x^2 - x) = -1
Simplify the numerator: (x^2 - x) / (x^2 - x) = -1
Divide the like terms: x / (x - 1) = -1
Multiply both sides by (x - 1) to eliminate the denominator: x = -1(x - 1)
Simplify the equation: x = -x + 1
Add x to both sides to simplify: 2x = 1
Divide by 2: x = 1/2
Therefore, the solution to the given equation is x = 1/2.
Given equation: (x / (x - 2)) + ((x - 1) / (x + 1)) = -1
Multiplying the first two terms: x * x = x^2
Multiplying the outside terms: x * 1 = x
Multiplying the inside terms: -2 * x = -2x
The equation now becomes: (x^2 + x - 2x) / (x^2 - x) = -1
Simplify the numerator: (x^2 - x) / (x^2 - x) = -1
Divide the like terms: x / (x - 1) = -1
Multiply both sides by (x - 1) to eliminate the denominator: x = -1(x - 1)
Simplify the equation: x = -x + 1
Add x to both sides to simplify: 2x = 1
Divide by 2: x = 1/2
Therefore, the solution to the given equation is x = 1/2.