We are given the equation:

$- \frac{x}{x+1} - \frac{5}{x+4} = \frac{1}{6}$

Step 1: Identify restrictions

The denominators $x+1$ and $x+4$ cannot be zero, because division by zero is undefined. Therefore, the restrictions on $x$ are: $x + 1 \neq 0 \quad \text{and} \quad x + 4 \neq 0$ $x \neq -1 \quad \text{and} \quad x \neq -4$ Thus, $x$ cannot be $-1$ or $-4$.

Step 2: Eliminate fractions by finding the common denominator

The least common denominator (LCD) of $x+1$, $x+4$, and 6 is $6(x+1)(x+4)$. Multiply both sides of the equation by this LCD to eliminate the denominators:

$6(x+1)(x+4) \left( - \frac{x}{x+1} - \frac{5}{x+4} \right) = 6(x+1)(x+4) \cdot \frac{1}{6}$

Step 3: Simplify each term

On the left-hand side:

• The first term simplifies to $-x \cdot 6(x+4)$, since the $x+1$ cancels out.
• The second term simplifies to $-5 \cdot 6(x+1)$, since the $x+4$ cancels out.

The right-hand side simplifies to $(x+1)(x+4)$.

So, we now have the equation:

$-6x(x+4) - 30(x+1) = (x+1)(x+4)$

Step 4: Expand both sides

Expand both sides of the equation:

$-6x(x+4) = -6x^2 - 24x$ $-30(x+1) = -30x - 30$ $(x+1)(x+4) = x^2 + 5x + 4$

Substitute these expressions back into the equation:

$-6x^2 - 24x - 30x - 30 = x^2 + 5x + 4$

Combine like terms on the left-hand side:

$-6x^2 - 54x - 30 = x^2 + 5x + 4$

Step 5: Move all terms to one side of the equation

Move all terms to the left-hand side:

$-6x^2 - 54x - 30 - x^2 - 5x - 4 = 0$ $-7x^2 - 59x - 34 = 0$

Step 6: Solve the quadratic equation

We now have the quadratic equation:

$-7x^2 - 59x - 34 = 0$

Use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = -7$, $b = -59$, and $c = -34$.

Step 7: Apply the quadratic formula

First, calculate the discriminant:

$b^2 - 4ac = (-59)^2 - 4(-7)(-34) = 3481 - 952 = 2529$

Now, apply the quadratic formula:

$x = \frac{-(-59) \pm \sqrt{2529}}{2(-7)} = \frac{59 \pm \sqrt{2529}}{-14}$

Approximating $\sqrt{2529}$:

$\sqrt{2529} \approx 50.29$

Thus:

$x = \frac{59 \pm 50.29}{-14}$

Step 8: Find the two solutions

Now compute the two possible values for $x$:

1. $x_1 = \frac{59 - 50.29}{-14} = \frac{8.71}{-14} \approx -0.62$
2. $x_2 = \frac{59 + 50.29}{-14} = \frac{109.29}{-14} \approx -7.81$

Step 9: Check for restrictions

Since neither $-0.62$ nor $-7.81$ equals $-1$ or $-4$, these values are valid solutions.

Final answer:

The solutions are: $x \approx -0.62 \quad \text{and} \quad x \approx -7.81$ The restrictions are $x \neq -1$ and $x \neq -4$.

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Would you like more details on any part of this process or further clarifications?

Here are 5 related questions:

1. How does the quadratic formula work in solving equations?
2. What is the significance of the discriminant in the quadratic formula?
3. Can you explain how to find the least common denominator for complex fractions?
4. How do restrictions in rational equations impact solutions?
5. Why do we need to check for excluded values after solving a rational equation?

Tip: Always check for restrictions when solving rational equations to avoid dividing by zero.