Simplify the difference. State every possible restictions of the variable. Show your work.
2x/x^2+x-6 - 4x/x^2-5x+6
To simplify the difference (2x/x^2 + x - 6) - (4x/x^2 - 5x + 6), we need to find a common denominator for both fractions. The denominators of both fractions are x^2 + x - 6 and x^2 - 5x + 6.
The first denominator can be factored as (x+3)(x-2), while the second denominator can be factored as (x-2)(x-3). Therefore, the common denominator is (x+3)(x-2)(x-3).
Using the common denominator, we can rewrite the expression as:
(2x(x-3) - 4x(x+3))/((x+3)(x-2)(x-3))
Next, we can distribute the 2x and -4x to get:
(2x^2 - 6x - 4x^2 - 12x)/((x+3)(x-2)(x-3))
Combining like terms in the numerator, we have:
(-2x^2 - 18x) / ((x+3)(x-2)(x-3))
The simplified difference is (-2x^2 - 18x) / ((x+3)(x-2)(x-3)).
Restrictions:
To find any restrictions on the variable "x", we need to examine the denominators. From the denominators (x+3), (x-2), and (x-3), we see that the value of "x" cannot be equal to -3, 2, or 3. Therefore, the restrictions on the variable "x" are x ≠ -3, x ≠ 2, and x ≠ 3.
To simplify the difference between the two expressions, we need to find a common denominator and then combine the fractions. Let's start by factoring the denominators.
First expression:
Denominator: x^2 + x - 6
Factoring: (x - 2)(x + 3)
Second expression:
Denominator: x^2 - 5x + 6
Factoring: (x - 2)(x - 3)
Now, we have the following fractions:
2x / (x - 2)(x + 3) - 4x / (x - 2)(x - 3)
The next step is to find a common denominator, which in this case is (x - 2)(x + 3)(x - 3). Let's rewrite the fractions with the common denominator:
[2x * (x - 3)] / [(x - 2)(x + 3)(x - 3)] - [4x * (x + 3)] / [(x - 2)(x + 3)(x - 3)]
Now, we can combine the fractions:
[2x(x - 3) - 4x(x + 3)] / [(x - 2)(x + 3)(x - 3)]
Expanding the terms:
[2x^2 - 6x - 4x^2 - 12x] / [(x - 2)(x + 3)(x - 3)]
Combining like terms:
[-2x^2 - 18x] / [(x - 2)(x + 3)(x - 3)]
Finally, simplifying the expression:
-2x(x + 9) / [(x - 2)(x + 3)(x - 3)]
Therefore, the simplified difference is -2x(x + 9) / [(x - 2)(x + 3)(x - 3)].
Restrictions on the variable can be found by considering the denominators in the expression. In this case, the variable cannot take the values x = 2, x = -3, or x = 3 since these values would result in a denominator of zero.
To simplify the difference between these two rational expressions, let's first factor both denominators:
x^2 + x - 6:
We need to find two numbers whose product is -6 and whose sum is 1 (the coefficient of x). The numbers that satisfy these conditions are -2 and 3:
So, x^2 + x - 6 = (x - 2)(x + 3).
x^2 - 5x + 6:
We need to find two numbers whose product is 6 and whose sum is -5 (the coefficient of x). The numbers that satisfy these conditions are -2 and -3:
So, x^2 - 5x + 6 = (x - 2)(x - 3).
Now, let's rewrite the expression:
2x/(x^2 + x - 6) - 4x/(x^2 - 5x + 6)
= 2x/((x - 2)(x + 3)) - 4x/((x - 2)(x - 3))
Next, we need to find the least common denominator (LCD) of the two fractions, which is (x - 2)(x + 3)(x - 3).
Now, we can rewrite the expression with the LCD:
= 2x * (x - 3)/(LCD) - 4x * (x + 3)/(LCD)
= (2x(x - 3) - 4x(x + 3))/(LCD)
= (2x^2 - 6x - 4x^2 - 12x)/(LCD)
= (-2x^2 - 18x)/(LCD)
= -2x(x + 9)/(LCD)
So, the simplified difference is -2x(x + 9)/(x - 2)(x + 3)(x - 3).
Restrictions on the variable:
To determine any restrictions for the variable x, we need to identify values that would make the denominator(s) equal to zero, as division by zero is undefined. From the factored denominators, we can see that x - 2, x + 3, and x - 3 cannot equal zero.
Therefore, the restrictions are x ≠ 2, x ≠ -3, and x ≠ 3.