Questions: Solve the following equation for (x). [ frac3x+3-frac2x-1=-frac1x^2+2 x-3 ] Provide your answer below: [ x= ]

$Solve the following equation for (x). [ frac3x+3-frac2x-1=-frac1x^2+2 x-3 ] Provide your answer below: [ x= ]$

Transcript text: Solve the following equation for $x$. \[ \frac{3}{x+3}-\frac{2}{x-1}=-\frac{1}{x^{2}+2 x-3} \] Provide your answer below: \[ x= \]

Solution

Solution Steps

To solve the given equation for $ x $, we need to find a common denominator for the fractions, combine them, and then solve the resulting polynomial equation. The common denominator will be $ (x+3)(x-1) $. After combining the fractions, we will set the numerator equal to zero and solve for $ x $.

Step 1: Identify the Common Denominator

To solve the equation \[ \frac{3}{x+3} - \frac{2}{x-1} = -\frac{1}{x^2 + 2x - 3}, \] we first identify the common denominator. The denominator $ x^2 + 2x - 3 $ can be factored as $ (x+3)(x-1) $.

Step 2: Combine the Fractions

Rewrite the equation with the common denominator: \[ \frac{3(x-1) - 2(x+3)}{(x+3)(x-1)} = -\frac{1}{(x+3)(x-1)}. \] Simplify the numerator: \[ 3(x-1) - 2(x+3) = 3x - 3 - 2x - 6 = x - 9. \] Thus, the equation becomes: \[ \frac{x - 9}{(x+3)(x-1)} = -\frac{1}{(x+3)(x-1)}. \]

Step 3: Set the Numerators Equal

Since the denominators are the same, we set the numerators equal: \[ x - 9 = -1. \]

Step 4: Solve for $ x $

Solve the equation: \[ x - 9 = -1 \implies x = 8. \]