Question

solve for all values of x:
$-6 - \frac{1}{x + 1} = \frac{x}{x + 1}$

Explanation:

Step1: Multiply both sides by \(x + 1\) (note \(x

eq - 1\))
Multiply each term by \(x + 1\) to eliminate the denominators: \(-6(x + 1)-1=x\)

Step2: Expand the left - hand side

Expand \(-6(x + 1)\): \(-6x-6 - 1=x\)
Simplify the left - hand side: \(-6x-7=x\)

Step3: Move all \(x\) terms to one side

Add \(6x\) to both sides: \(-7 = x+6x\)
Combine like terms: \(-7 = 7x\)

Step4: Solve for \(x\)

Divide both sides by 7: \(x=-1\)
But we need to check if \(x = - 1\) is valid. When \(x=-1\), the denominator \(x + 1=-1 + 1 = 0\), so \(x=-1\) makes the original equation undefined. So there is no solution.

Answer:

No solution