find x

4x-5/1-5x = x+1/x+7

please can you show me the method
thanks

Sure! To solve this equation for x, we'll need to follow a step-by-step method. Here's how:

Step 1: Clear the fractions by multiplying both sides of the equation by the common denominator of all the fractions, which is (x+7)(x+1):

[(4x - 5) / (1 - 5x)] * (x+7)(x+1) = (x+1) / (x+7)

Step 2: Distribute and simplify:

(4x - 5)(x+7)(x+1) = (x+1)^2

Step 3: Expand and simplify both sides of the equation:

(4x^2 + 28x - 5x - 35)(x+1) = x^2 + 2x + 1

(4x^2 + 23x - 35)(x+1) = x^2 + 2x + 1

Step 4: Expand the equation:

4x^3 + 23x^2 - 35x + 4x^2 + 23x - 35 = x^2 + 2x + 1

4x^3 + (23x^2 + 4x^2) + (23x + 23x) + (-35 - 35) = x^2 + 2x + 1

4x^3 + 27x^2 + 46x - 70 = x^2 + 2x + 1

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

4x^3 + 27x^2 + 46x - x^2 - 2x - 1 - 70 = 0

4x^3 + 26x^2 + 44x - 71 = 0

Step 6: Solve the cubic equation. Unfortunately, there is no general formula to solve cubic equations, but you can use numerical methods like the Rational Root Theorem, synthetic division, or a graphing calculator to estimate the solutions.

Alternatively, you can use numerical methods such as using an online equation solver or computer software to find the approximate solutions for x.

I hope this helps! Let me know if you have any further questions.