Simplify the difference. State every possible restictions of the variable. Show your work.
4x/x^2+2x-3 - 6x/x^2-4x+3
To simplify the difference of the given rational expressions, we start by factoring the denominators of each expression.
For the first expression, x^2 + 2x - 3, we can factor it as (x + 3)(x - 1).
For the second expression, x^2 - 4x + 3, we can factor it as (x - 1)(x - 3).
Now we can express the given expressions with their factored denominators:
(4x)/(x^2 + 2x - 3) - (6x)/(x^2 - 4x + 3)
Next, we need to find the least common denominator (LCD) of the two expressions, which is (x + 3)(x - 1)(x - 3).
To make the denominators of the expressions equal to the LCD, we multiply the numerator and denominator of the first expression by (x - 1)(x - 3), and we multiply the numerator and denominator of the second expression by (x + 3)(x - 3):
[(4x)(x - 1)(x - 3)]/[(x + 3)(x - 1)(x - 3)] - [(6x)(x + 3)(x - 3)]/[(x + 3)(x - 1)(x - 3)]
Now that the denominators are the same, we can combine the numerators:
[4x(x - 1)(x - 3) - 6x(x + 3)(x - 3)]/[(x + 3)(x - 1)(x - 3)]
Simplifying the numerator:
[4x(x^2 - 4x + 3) - 6x(x^2 - 9)]/[(x + 3)(x - 1)(x - 3)]
(4x^3 - 16x^2 + 12x - 6x^3 + 54x)/[(x + 3)(x - 1)(x - 3)]
(-2x^3 - 16x^2 + 66x)/[(x + 3)(x - 1)(x - 3)]
Now we have the simplified expression:
(-2x^3 - 16x^2 + 66x)/[(x + 3)(x - 1)(x - 3)]
The possible restrictions on the variable x are when the denominator would be equal to zero:
x + 3 = 0 --> x = -3
x - 1 = 0 --> x = 1
x - 3 = 0 --> x = 3
Thus, the possible restrictions on the variable x are x ≠ -3, x ≠ 1, and x ≠ 3.
To simplify the difference (4x/x^2 + 2x - 3) - (6x/x^2 - 4x + 3), we need to find a common denominator for the two fractions.
First, let's factor the denominators:
x^2 + 2x - 3 = (x + 3)(x - 1)
x^2 - 4x + 3 = (x - 3)(x - 1)
Now we can find the common denominator by taking the product of the two factored denominators:
(x + 3)(x - 1)(x - 3)
Next, we need to rewrite both fractions with the common denominator:
(4x(x - 3))/(x + 3)(x - 1)(x - 3) - (6x(x + 3))/(x + 3)(x - 1)(x - 3)
Now we can combine the fractions by subtracting the numerators:
(4x(x - 3) - 6x(x + 3))/(x + 3)(x - 1)(x - 3)
Expanding the expressions in the numerator:
(4x^2 - 12x - 6x^2 - 18x)/(x + 3)(x - 1)(x - 3)
Combining like terms in the numerator:
(-2x^2 - 30x)/(x + 3)(x - 1)(x - 3)
Now, let's analyze the possible restrictions of the variable:
1. The denominator cannot equal zero to avoid division by zero. So we set each factor equal to zero and solve:
x + 3 ≠ 0 => x ≠ -3
x - 1 ≠ 0 => x ≠ 1
x - 3 ≠ 0 => x ≠ 3
Therefore, the restrictions on the variable x are x ≠ -3, x ≠ 1, and x ≠ 3.
To recap, we simplified the difference to:
(-2x^2 - 30x)/(x + 3)(x - 1)(x - 3)
with the restrictions x ≠ -3, x ≠ 1, and x ≠ 3.
To simplify the given expression, we need to find a common denominator and combine the fractions. Let's break it down step by step.
Step 1: Factorize the denominators:
x^2 + 2x - 3 factors to (x + 3)(x - 1)
x^2 - 4x + 3 factors to (x - 3)(x - 1)
Step 2: Set up the common denominator:
The common denominator will be (x + 3)(x - 1).
Step 3: Rewrite the fractions with the common denominator:
4x/(x^2 + 2x - 3) can be written as 4x/[(x + 3)(x - 1)]
6x/(x^2 - 4x + 3) can be written as 6x/[(x - 3)(x - 1)]
Step 4: Combine the fractions:
The expression becomes:
(4x/[(x + 3)(x - 1)]) - (6x/[(x - 3)(x - 1)])
Step 5: Determine the difference of the fractions:
To find the difference of the fractions, we need a common denominator:
[(4x)(x - 3)] - [(6x)(x + 3)]
= (4x^2 - 12x) - (6x^2 + 18x)
= 4x^2 - 12x - 6x^2 - 18x
= -2x^2 - 30x
Therefore, the simplified expression is -2x^2 - 30x.
Restrictions on the variable:
To determine the restrictions, we need to look at the factors in the denominators: (x + 3)(x - 1) and (x - 3)(x - 1).
Since we have division by these factors, we need to ensure that they are not equal to zero. Therefore, the restrictions on x are:
x ≠ -3, 1 (from the (x + 3)(x - 1))
x ≠ 3, 1 (from the (x - 3)(x - 1))
So, the variable x cannot take the values -3, 1, 3.