solve for x, stating restrictions:  - x/(x+1) - 5/(x+4) = 1/6

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 \(x \neq -4\).

---

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.

solve for x, stating restrictions: - x/(x+1) - 5/(x+4) = 1/6

Learn how to solve the rational equation - x/(x+1) - 5/(x+4) = 1/6, including restrictions and solving using the quadratic formula. This problem covers important algebraic concepts such as LCD, restrictions, and the quadratic formula. Suitable for students in Grades 9-12.

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