Questions: Solve for all values of x. (x+1)/(x-6)=(-4)/x

Transcript text: Solve for all values of $x$. \[ \frac{x+1}{x-6}=\frac{-4}{x} \]

Solution

Step 1: Cross-Multiply the Equation

We start with the equation \[ \frac{x+1}{x-6} = \frac{-4}{x}. \] By cross-multiplying, we obtain: \[ (x + 1) \cdot x = -4 \cdot (x - 6). \]

Step 2: Simplify the Equation

Expanding both sides gives us: \[ x^2 + x = -4x + 24. \] Rearranging this leads to: \[ x^2 + 5x - 24 = 0. \]

Step 3: Solve the Quadratic Equation

We can factor or use the quadratic formula to solve for $x$. The solutions to the equation are: \[ x = -8 \quad \text{and} \quad x = 3. \]

The values of $x$ that satisfy the equation are \[ \boxed{x = -8} \quad \text{and} \quad \boxed{x = 3}. \]

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