Questions: (x+1)/(x^2-2x-3)-(1)/(x^2+x)-(3)/(x^2-3x)=(1)/(x+1), x ≠ 0, x ≠ 3

Transcript text: b) $\frac{x+1}{x^{2}-2 x-3}-\frac{1}{x^{2}+x}-\frac{3}{x^{2}-3 x}=\frac{1}{x+1}, x \neq 0, x \neq 3$

Solution

Solution Steps

To solve the given equation, we need to find a common denominator for the fractions on the left-hand side and then simplify the equation. After simplifying, we can equate the numerators and solve for $ x $.

Step 1: Identify the Common Denominator

To solve the equation: \[ \frac{x+1}{x^2 - 2x - 3} - \frac{1}{x^2 + x} - \frac{3}{x^2 - 3x} = \frac{1}{x+1} \] we first identify the common denominator for the fractions on the left-hand side. The denominators are: \[ x^2 - 2x - 3, \quad x^2 + x, \quad x^2 - 3x \]

Step 2: Factorize the Denominators

Factorize each denominator: \[ x^2 - 2x - 3 = (x-3)(x+1) \] \[ x^2 + x = x(x+1) \] \[ x^2 - 3x = x(x-3) \]

Step 3: Combine the Fractions

Combine the fractions over the common denominator $(x-3)(x+1)x$: \[ \frac{(x+1)x(x-3) - (x-3)(x+1) - 3x(x+1)}{(x-3)(x+1)x} = \frac{1}{x+1} \]

Step 4: Simplify the Numerator

Simplify the numerator: \[ (x+1)x(x-3) - (x-3)(x+1) - 3x(x+1) = 0 \]

Step 5: Solve the Simplified Equation

Equate the simplified numerator to zero and solve for $x$: \[ (x+1)x(x-3) - (x-3)(x+1) - 3x(x+1) = 0 \]

Step 6: Filter Out Invalid Solutions

The solutions to the equation are filtered to exclude $x = 0$ and $x = 3$.