# Quadratic inequalities involving two solutions

I have a quadratic expression, which I have factored to correctly be: $(x-9)(x-2) > 0$

However, I don't know how to determine the two values of X after this, the correct answers are x < 2, x > 9, however, is this because x is smaller, and 9 is larger, or is their a specific algebraic step in solving this?

• For two numbers to give positive product, either both numbers are positive, or both numbers are negative. – peterwhy May 31 '14 at 13:38
• That was the formal way of writing solution; I usually think faster in the following way. If you are familiar enough with quadratic curves, you know $y=(x-9)(x-2)$ gives a U-shaped curve. Knowing $x=2$ and $9$ are roots, which part can then give $y>0$? Both ends, $x<2$ or $x>9$, instead of $2<x<9$. – peterwhy May 31 '14 at 13:46

If both are positive, then $x-9 > 0$ and $x-2 > 0$; from this you can see that $x >9$ and $x >2$. But the latter condition is redundant, since if $x >9$ then certainly $x >2$. Therefore, if both factors are positive, then $x >9$.
If both factors are negative, then following a similar chain of logic, you will get that $x <2$.
Thus either $x>9$ or $x<2$.
• As an alternative to this algebraic approach, you could try a graphical approach. The left-hand side is a parabola with $x$-intercepts at $x=2$ and $x=9$. When is the parabola above the $y$-axis? – Théophile May 31 '14 at 13:43
• @mahat Yes, that is correct. To make sure you understand this kind of reasoning, consider what happens if you reverse the inequality in the question: $(x-9)(x-2) < 0$. Then one product must be positive and one must be negative; try working out the logic from there. – Théophile May 31 '14 at 13:53