Inequalities

At a glance 13 formulas

1. 1Multiply by a negative$x < y,\ k < 0 \Rightarrow kx > ky$$-3x \le 12 \Rightarrow x \ge -4$
2. 2Adding inequalities$a < b,\ c < d \Rightarrow a + c < b + d$$\text{never subtract or divide two inequalities}$
3. 3Reciprocals$0 < a < b \Rightarrow \dfrac{1}{a} > \dfrac{1}{b}$$\text{same when } a < b < 0$
4. 4Counting integers$a \le x \le b:\ b - a + 1 \text{ integers}$$a < x < b:\ b - a - 1 \text{ integers } (a, b \text{ integers})$
5. 5Quadratic inequality$(x - a)(x - b) < 0 \iff a < x < b$$(x - a)(x - b) > 0 \iff x < a \text{ or } x > b \ (a < b)$
6. 6Real rootsFor $ax^2 + bx + c = 0$ to have a real $x$, $D = b^2 - 4ac \ge 0$; if $a > 0$ and $D < 0$, $ax^2 + bx + c$ is positive for every $x$.
7. 7Wavy curveFactor fully and mark the roots. The sign is + to the right of the largest root and flips at each root, except at a squared factor, where it stays.
8. 8x in the denominatorNever cross-multiply by an expression whose sign you don't know: move everything to one side and sign-chart it. A root of the denominator is never in the answer.
9. 9Size of powers$0 < x < 1 \Rightarrow x^3 < x^2 < x < \sqrt{x}$$x > 1 \Rightarrow \sqrt{x} < x < x^2 < x^3$
10. 10Squaring a rangeIf the range of $x$ crosses 0, $x^2$ starts at 0 and goes up to the bigger end squared: $-3 < x < 2$ gives $0 \le x^2 < 9$.
11. 11Powers of one base$a > 1:\ a^m > a^n \iff m > n$$0 < a < 1:\ a^m > a^n \iff m < n$
12. 12Test three valuesFor 'which must be true', try a negative number, a fraction between 0 and 1, and a number above 1. One failure kills a statement.
13. 13Sum of squares$a^2 + b^2 + c^2 \ge ab + bc + ca$$ab + bc + ca \ge -\tfrac{1}{2}(a^2 + b^2 + c^2)$