# Thread: inequalities of rational expressions

1. ## inequalities of rational expressions

referring to the photo attached, can anyone expalin and give suitable numerical example please? i couldnt understand the notes.

2. ## Re: inequalities of rational expressions

$\displaystyle \dfrac{5}{-4} < \dfrac{5}{4}$

If you were to cross-multiply, but keep the inequality as less than, you would get

$\displaystyle 20 = 5(4) < 5(-4) = -20$, which is clearly false.

Now, if you have an expression like

$\displaystyle \dfrac{f(x)}{g(x)} < \dfrac{h(x)}{j(x)}$, you must be certain of whether it is positive or negative. So, you can say the following:

$\displaystyle f(x)j(x) < g(x)h(x)$ if $\displaystyle g(x)j(x)>0$

$\displaystyle f(x)j(x) > g(x)h(x)$ if $\displaystyle g(x)j(x)<0$

Example 1:

$\displaystyle \dfrac{5}{(x+1)^2} < \dfrac{7x}{5}$

Since $\displaystyle 5(x+1)^2>0$ for all $\displaystyle x$, you know $\displaystyle 5(5) < 7x(x+1)^2$

Example 2:

$\displaystyle \dfrac{5}{-4} < \dfrac{7x^3}{1-(x+2)^2}$

Since $\displaystyle -4[1-(x+2)^2] = 4[(x+2)^2-1] \ge 4[(0+2)^2-1] = 4(4-1) = 12>0$, you know that $\displaystyle 5[1-(x+2)^2] < -7x^3(-4)$

Example 3:

$\displaystyle \dfrac{5}{-4} < \dfrac{7x^3}{(x+1)^2}$

Since $\displaystyle -4(x+1)^2 < 0$ for all $\displaystyle x$, you know $\displaystyle 5(x+1)^2 > 7x^3(-4)$ (note the inequality changed direction because of the single negative).