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Sign charts

To solve a polynomial inequality like $(x-2)(x+3) < 0$ , find the roots of the polynomial (here, $x=2$ and $x=-3$ ), mark them on a number line, and test one point in each resulting interval to see whether the expression is positive or negative there. The sign only changes when you cross a root of odd multiplicity — so once you know the sign in one interval, the rest follow by alternation (for distinct linear factors).

The AM-GM inequality

For any nonnegative real numbers, the arithmetic mean is always at least the geometric mean:

\dfrac{a+b}{2} \geq \sqrt{ab}, \qquad\text{equality iff } a=b

More generally, for $n$ nonnegative reals:

\dfrac{a_1+a_2+\cdots+a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}

AM-GM is the go-to tool whenever a problem asks you to minimize a sum given a fixed product (or maximize a product given a fixed sum) — and it requires the terms to be nonnegative, a condition worth checking before you reach for it.

Worked example

Problem: For $x > 0$ , find the minimum value of $x + \dfrac{9}{x}$ .

Solution: Apply AM-GM to the two positive terms $x$ and $\tfrac9x$ :

\dfrac{x + \tfrac9x}{2} \geq \sqrt{x \cdot \tfrac9x} = \sqrt 9 = 3 \;\Longrightarrow\; x + \dfrac9x \geq 6

Equality holds when the two terms are equal: $x = \tfrac9x \Rightarrow x^2 = 9 \Rightarrow x=3$ (taking the positive root, since $x>0$ ). So the minimum value is $6$ , achieved at $x=3$ .

Practice problems

1.Solve $(x-1)(x-4) > 0$ and write the answer in interval notation.

2.For positive reals $a,b$ with $ab=16$ , find the minimum possible value of $a+b$ .

3.Solve $x^2 - 5x + 6 \leq 0$ .

4.For $x>0$ , find the minimum value of $4x + \dfrac{1}{x}$ .

Inequalities & Optimization

Sign charts

The AM-GM inequality

Practice problems