Hi, I am looking at solving a rational inequality algebraically and the textbook says to multiply by the denominator squared, is this because squaring would remove the effect of a negative value of x and hence a change in sign. Thanks
Yes, that's essentially the reason. More precisely, it's because the denominator may be negative for some values of x. However, if your denominator is a perfect square, you could simply multiply both sides by the denominator without further squaring it.
There are other methods which are also designed to avoid multiplying by the denominator, when it's not clearly non-negative.