Simplify the difference. State any restrictions on the variable.
3x/x^2+x-6 - 6x/x^2-5x+6
close, but no cigar...
-3x(x + 9)/((x + 3)(x - 2)(x - 3))
Apologies for the mistake. You are correct, the simplified difference should be:
-3x(x + 9)/((x + 3)(x - 2)(x - 3))
Thank you for catching that.
To simplify the difference (3x/x^2+x-6) - (6x/x^2-5x+6), we need to find a common denominator for the two fractions.
First, let's factor the denominators:
x^2+x-6 = (x+3)(x-2)
x^2-5x+6 = (x-2)(x-3)
Now, we can rewrite the expression with the common denominator:
(3x*(x-2))/((x+3)(x-2)) - (6x*(x+3))/((x-2)(x-3))
Next, we can combine the numerators:
(3x*(x-2) - 6x*(x+3))/((x+3)(x-2))
Expanding the terms:
(3x^2 - 6x - 6x^2 - 18x)/((x+3)(x-2))
Combining like terms:
(-3x^2 - 24x)/((x+3)(x-2))
Finally, we can simplify further if possible:
-3x(x+8)/((x+3)(x-2))
Therefore, the simplified difference is -3x(x+8)/((x+3)(x-2)). As for restrictions on the variable, we need to check for any values that would make the denominator zero since division by zero is undefined.
The restrictions would be when:
(x+3) = 0, which means x = -3
(x-2) = 0, which means x = 2
Therefore, the restrictions on the variable are x ≠ -3 and x ≠ 2.
To simplify the difference (or subtraction) of the given rational expressions, we need to combine them into a single fraction.
First, let's find the least common denominator (LCD) of the two expressions, which is the least common multiple of the denominators (x^2 + x - 6) and (x^2 - 5x + 6).
Factoring the denominators, we have:
x^2 + x - 6 = (x + 3)(x - 2)
x^2 - 5x + 6 = (x - 2)(x - 3)
The LCD is the product of the unique factors raised to their highest powers:
LCD = (x + 3)(x - 2)(x - 3).
To make the denominators of the two expressions equal to the LCD, we need to multiply the first expression by (x - 2)/(x - 2) and the second expression by (x + 3)/(x + 3):
3x/(x^2 + x - 6) - 6x/(x^2 - 5x + 6)
= 3x(x - 2)/[(x + 3)(x - 2)] - 6x(x + 3)/[(x - 2)(x - 3)]
Now that the denominators are the same, we can subtract the numerators:
= [3x(x - 2) - 6x(x + 3)]/[(x + 3)(x - 2)]
Simplifying the numerator:
= (3x^2 - 6x^2 + 12x - 18x)/[(x + 3)(x - 2)]
= (-3x^2 - 6x)/[(x + 3)(x - 2)]
= -3x(x + 2)/[(x + 3)(x - 2)]
Therefore, the simplified difference of the two rational expressions is -3x(x + 2)/[(x + 3)(x - 2)].
As for the restrictions on the variable, we need to consider any values of x that would make the denominator equal to zero. In this case, the restrictions are:
(x + 3)(x - 2) ≠ 0
Therefore, x cannot be -3 or 2. These values would result in a zero denominator, which is undefined in rational expressions. So the restrictions on the variable are x ≠ -3 and x ≠ 2.
To simplify the difference, we need to first find a common denominator. The denominators of the two fractions are (x^2 + x - 6) and (x^2 - 5x + 6).
To find the common denominator, we need to factor both denominators:
x^2 + x - 6 = (x + 3)(x - 2)
x^2 - 5x + 6 = (x - 2)(x - 3)
The common denominator is (x + 3)(x - 2)(x - 3).
Now we can rewrite the fractions with the common denominator:
(3x/(x^2 + x - 6)) - (6x/(x^2 - 5x + 6))
Multiplying the first fraction by (x - 2)/(x - 2) and the second fraction by (x + 3)/(x + 3), we get:
(3x(x - 2)/((x^2 + x - 6)(x - 2))) - (6x(x + 3)/((x^2 - 5x + 6)(x + 3)))
Simplifying further:
(3x^2 - 6x)/((x + 3)(x - 2)(x - 3)) - (6x^2 + 18x)/((x + 3)(x - 2)(x - 3))
Now, we can subtract the two fractions:
(3x^2 - 6x - 6x^2 - 18x)/((x + 3)(x - 2)(x - 3))
Simplifying the numerator:
(-3x^2 - 24x)/((x + 3)(x - 2)(x - 3))
Finally, we can simplify further:
-3x(x + 8)/((x + 3)(x - 2)(x - 3))
There are no restrictions on the variable.