# How do you divide an inequality by another inequality?

The question I'm having trouble with is follows:

Suppose you have already proved the proposition that, “If $$a$$ and $$b$$ are nonnegative real numbers, then $$\frac{a + b}{2} ≥ \sqrt{ab}$$.”

a. Explain how you could use this proposition to prove that if a and b are real numbers satisfying the property that $$b ≥ 2|a|$$, then $$b ≥ \sqrt{b^2 − 4a^2}$$. Be careful how you match up notation.

b. Use the foregoing proposition and part (a) to prove that if $$a$$ and $$b$$ are real numbers with $$a < 0$$ and $$b ≥ 2|a|$$, then one of the roots of the equation $$ax^2 + bx + a = 0$$ is $$≤ -\frac{b}{a}.$$

I'm stuck on the solution to $$b$$. (sadly), because the explanation involves dividing an inequality by another inequality and I have no idea how it works.

Here is the solution: solution screenshot

Between $$A_1$$ and $$A_2$$ in solution to b., it says "Subtracting $$2b$$ from both sides of $$A_1$$ and then dividing by $$2a <0$$ (from the hypothesis)". I don't know how dividing by $$2a <0$$ works. I mean, I know how to divide an equation by another equation in systems of equations, but oh god what is this. Please help me

• Please try to use link LaTeX next time. – Jaemin Kim Oct 25 '19 at 4:07

You shouldn't read that line as "divide by the inequality $$2a<0$$." Rather, read it as "divide both sides by $$2a$$, which is negative (so dividing by $$2a$$ flips the direction of the inequality)."
In some situations, it may be possible to "divide an inequality by another inequality." If $$a > b$$ and $$0 < c < d$$, then $$\frac{a}{c}>\frac{b}{d}$$, and you could perhaps think of this conclusion as being reached by dividing the inequality $$a>b$$ by the inequality $$c < d$$. But even in this case it would be clearer to split the argument up into two steps: first note that $$\frac{a}{c}>\frac{b}{c}$$, and then that $$\frac{b}{c}>\frac{b}{d}$$.
Well, think of it as simply dividing by a negative quantity. That's the idea you want to get from seeing $$2a<0.$$ And actually, it's only in such cases (when one side of a strict inequality is $$0$$) that you can divide by such inequalities.
In sum, what we've done is to divide by $$2a,$$ and this reverses the order accordingly.