# Inequalities

Real numbers: $$a$$, $$b$$, $$c$$, $$d$$, $$x$$, $$m$$, $$n$$
Real positive numbers: $${a_1}$$, $${a_2}$$, …, $${a_n}$$
1. Inequalities and intervals of the number line
2. Strict inequalities
The notation $$a \lt b$$ means that “$$a$$ is less than $$b$$”,
The notation $$a \gt b$$ means that “$$a$$ is greater than $$b$$”.
3. Non-strict inequalities
The notation $$a \le b$$ means that “$$a$$ is less than or equal to $$b$$”,
The notation $$a \ge b$$ means that “$$a$$ is greater than or equal to $$b$$”.
4. If $$a \gt b$$, then $$b \lt a$$.
5. If $$a \gt b$$, then $$a – b \gt 0$$ or equivalently $$b – a \lt 0$$.
6. Transitivity property
If$$a \gt b$$ and $$b \gt c$$, then $$a \gt c$$.
7. Adding the same number to the both sides of an inequality does not change the sign of the inequality:
If $$a \gt b$$, then $$a + c \gt b + c$$.
8. Moving any term of an inequality from one side to another and changing its sign results in an equivalent inequality:
If $$a + b \gt c$$, then $$a \gt c – b$$.
9. If $$a \gt b$$ and $$c \gt d$$, then $$a + c \gt b + d$$.
10. If $$a \gt b$$ and $$c \gt d$$, then $$a – d \gt b – c$$.
11. Multiplying both sides of an inequality by the same positive number does not change the sign of the inequality:
If $$a \gt b$$ and $$m \gt 0$$, then $$ma \gt mb$$.
12. Dividing both sides of an inequality by the same positive number does not change the sign of the inequality:
If $$a \gt b$$ and $$m \gt 0$$, then $$a/m > b/m$$.
13. Multiplying both sides of an inequality by the same negative number inverts the inequality:
If $$a \gt b$$ and $$m \lt 0$$, then $$ma \lt mb$$.
14. Dividing both sides of an inequality by the same negative number inverts the inequality:
If $$a \gt b$$ and $$m \lt 0$$, then $$a/m \lt b/m$$.
15. If $$a \gt b \gt 0$$, then $$1/b \gt 1/a$$.
16. Multiplication of inequalities
If $$a \gt b \gt 0$$ and $$c \gt d \gt 0$$, then $$ac \gt bd$$.
17. Division of inequalities
If $$a \ge b \gt 0$$ and $$c \gt d \gt 0$$, then $$a/d \gt b/c$$.
18. Raising an inequality to a positive power
If $$a \gt b \gt 0$$ and $$n \gt 0$$, then $${a^n} \gt {b^n}$$.
19. Raising an inequality to a negative power
If $$a \gt b \gt 0$$ and $$n \lt 0$$, then $${a^n} \lt {b^n}$$.
20. $$N$$th root of an inequality
If $$a \gt b \gt 0$$ and $$n \gt 0$$, then $$\sqrt[\large n\normalsize]{a} \gt \sqrt[\large n\normalsize]{b}$$.
21. $$a + \large\frac{1}{a}\normalsize \ge 2\;$$ $$\left( {a \gt 0} \right)$$
The equality holds only if $$a = 1$$.
22. Cauchy’s inequality of arithmetic and geometric means
$$\sqrt {ab} \le \left( {a + b} \right)/2,$$ where $$a \gt 0,$$ $$b \gt 0$$.
The equality holds only when $$a = b$$.
23. Cauchy’s inequality of arithmetic and geometric means (the case of several variables)
$$\sqrt[n]{{{a_1}{a_2} \cdots {a_n}}} \le$$ $${\large\frac{{{a_1} + {a_2} + \ldots + {a_n}}}{n}\normalsize},$$ where $${a_1},{a_2}, \ldots ,{a_n} \gt 0$$.
24. Linear inequality (case $$a \gt 0$$)
If $$ax + b \gt 0$$ and $$a \gt 0$$, then $$x \gt -b/a$$.
25. Linear inequality (case $$a \lt 0$$) If $$ax + b \gt 0$$ and $$a \lt 0$$, then $$x \lt -b/a$$.
26. Quadratic inequality $$a{x^2} + bx + c \gt 0$$
27. $$\left| {a + b} \right| \le \left| a \right| + \left| b \right|$$
28. If $$\left| x \right| \lt a$$, then $$-a \lt x \lt a,$$ where $$a \gt 0$$.
29. If $$\left| x \right| \gt a$$, then $$x \lt -a$$ and $$x \gt a,$$ where $$a \gt 0$$.
30. If $${x^2} \lt a$$, then $$\left| x \right| \lt \sqrt a$$, where $$a \gt 0$$.
31. If $${x^2} \gt a$$, then $$\left| x \right| \gt \sqrt a$$, where $$a \gt 0$$.
32. $${\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \gt 0,\;$$ $$\Leftrightarrow \;f\left( x \right)g\left( x \right) \gt 0\;$$ $$\Leftrightarrow \;\begin{cases} {f\left( x \right)} \gt 0 \\ {g\left( x \right)} \gt 0 \end{cases}\;$$ $$\text { or }\; \begin{cases} {f\left( x \right)} \lt 0 \\ {g\left( x \right)} \lt 0 \end{cases}.$$
33. $${\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \lt 0,\;$$ $$\Leftrightarrow \;f\left( x \right)g\left( x \right) \lt 0\;$$ $$\Leftrightarrow \;\begin{cases} {f\left( x \right)} \gt 0 \\ {g\left( x \right)} \lt 0 \end{cases}\;$$ $$\text { or }\; \begin{cases} {f\left( x \right)} \lt 0 \\ {g\left( x \right)} \gt 0 \end{cases}.$$
34. $${\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \ge 0,\;$$ $$\Leftrightarrow \; \begin{cases} {f\left( x \right) g\left( x \right)} \ge 0 \\ {g\left( x \right)} \ne 0 \end{cases}\;$$ $$\Leftrightarrow \;\begin{cases} {f\left( x \right)} \ge 0 \\ {g\left( x \right)} \gt 0 \end{cases}\;$$ $$\text { or }\; \begin{cases} {f\left( x \right)} \le 0 \\ {g\left( x \right)} \lt 0 \end{cases}.$$
35. $${\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \le 0,\;$$ $$\Leftrightarrow \; \begin{cases} {f\left( x \right) g\left( x \right)} \le 0 \\ {g\left( x \right)} \ne 0 \end{cases}\;$$ $$\Leftrightarrow \;\begin{cases} {f\left( x \right)} \le 0 \\ {g\left( x \right)} \gt 0 \end{cases}\;$$ $$\text { or }\; \begin{cases} {f\left( x \right)} \ge 0 \\ {g\left( x \right)} \lt 0 \end{cases}.$$