Simplify each expression. State all non-permissible values.
š„+2 / š„ā1 ā š„ā3 / 1āš„
To simplify the expression, we need to combine the fractions.
The least common denominator (LCD) is (x - 1)(1 - x).
Multiplying the first fraction by (1 - x) and the second fraction by (x - 1), we get:
(x + 2)(1 - x) / [(x - 1)(1 - x)] - (x - 3)(x - 1) / [(x - 1)(1 - x)]
Expanding and simplifying each numerator:
(x + 2)(1 - x) = -x^2 - x + 2
(x - 3)(x - 1) = x^2 - 4x + 3
The expression becomes:
(-x^2 - x + 2) / [(x - 1)(1 - x)] - (x^2 - 4x + 3) / [(x - 1)(1 - x)]
Factoring out -1 from the first numerator:
-1(x^2 + x - 2) / [(x - 1)(1 - x)] - (x^2 - 4x + 3) / [(x - 1)(1 - x)]
Combining the numerators:
[-1(x^2 + x - 2) - (x^2 - 4x + 3)] / [(x - 1)(1 - x)]
Simplifying the numerator:
[-x^2 - x + 2 - x^2 + 4x - 3] / [(x - 1)(1 - x)]
[-2x^2 + 3x - 1] / [(x - 1)(1 - x)]
Now we can simplify further by dividing both numerator and denominator by -1:
[2x^2 - 3x + 1] / [(1 - x)(x - 1)]
Simplifying the denominator:
[(x - 1)(x - 1)] / [(1 - x)(x - 1)]
Cancelling out common factors in the numerator and denominator:
(x - 1) / (1 - x)
To find the non-permissible values, we need to consider the values that would make the denominator zero. In this case, when (1 - x) would be equal to zero, i.e., when x = 1.
Therefore, the non-permissible value of x is 1.