How would go about finding the set of values of x for which x+4 > 4 / (x+1)?

Whenever you see a problem involving an inequality (greater than or less than sign) it is really important to pause for a second before you go breaking any rules of mathematics. Inequalities do not always behave the same way as equalities (equals sign), especially when negative numbers are involved. Take this simple inequality: 1 < 2. We know that this is true but if we multiply both sides by a negative number such as -1 we get the new statement -1 < -2 which isn't true at all. Since (x+1) could be negative (if x is less than -1) we can't multiply both sides by it however we can multiply both sides by (x+1)^2 since that will always be greater than or equal to zero. We do just that:

x+4 > 4 / (x+1) (x+4)(x+1)^2 > 4(x+1) [Since one of the (x-1)'s on the numerator of right-hand side cancels with the one on the denominator]

(x+4)(x^2 + 2x + 1) > 4x + 4 [We start expanding out the brackets...]

x(x^2 + 2x + 1) + 4(x^2 + 2x + 1) > 4x + 4

x^3 + 2x^2 + x + 4x^2 + 8x + 4 > 4x + 4

x^3 + 6x^2 + 5x > 0 [...and collect all terms on one side of the inequality]

It is really tempting to divide by x at this point since every term a multiple of x but beware. x could be zero and division by zero is never allowed so instead we solve the inequality by factorising it:

x(x^2 + 6x + 5) > 0

x(x+1)(x+5) > 0

We can now sketch a graph of x(x+1)(x+5) by using the fact that it evaluates to zero for x = 0, -1 and -5 and using the typical shape of a cubic equation where the coefficient of x^3 is positive (it's 1 in this case). We then shade in the regions that are greater than zero (above the x-axis) and turn these into the range of values of x that satisfy the inequality:

-5 < x < -1 or x > 0