Basic Mathematics: Lecture 03 - Inequalities and Algebraic Identities

Practice fill-in-the-blank flashcards covering algebraic identities, inequality rules, and the wavy curve method from Lecture 03 of Basic Mathematics.

For any three variables, the value of the algebraic expression $\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{(x - y)(y - z)(z - x)}$ is __________.

$$3$$

If $x^{1/3} + y^{1/3} + z^{1/3} = 0$, then $(x + y + z)^3$ is equal to __________.

$$27xyz$$

If $x, y, z$ are all distinct real numbers, the sum of squares $\frac{1}{(x-y)^2} + \frac{1}{(y-z)^2} + \frac{1}{(z-x)^2}$ can be written as __________ because the sum of pairwise cross-terms equals zero.

$$\left(\frac{1}{x-y} + \frac{1}{y-z} + \frac{1}{z-x}\right)^2$$

If $a + b + c = 6$, $a^2 + b^2 + c^2 = 14$, and $a^3 + b^3 + c^3 = 36$, then the value of $3\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)$ is __________.

$$5.5$$

If $x = 2^{1/3} - 2^{-1/3}$, then the expression $2x^3 + 6x$ evaluates to __________.

$$3$$

If $x = 3 + 3^{1/3} - 3^{2/3}$, the value of $x^3 - 9x^2 + 36x$ is __________.

$$48$$

According to the wavy curve method, the complete solution set of $(x - 1)^2(x - 5) \ge 0$ is __________.

$$[5, \infty) \cup \{1\}$$

The solution set for the inequality $\frac{x - 3}{x + 2} \le 0$ is __________.

$$(-2, 3]$$

The complete real solution set of the rational inequality $\frac{x(x - 2)^2}{(x + 3)^3} \le 0$ is __________.

$$(-3, 0] \cup \{2\}$$

The solution set satisfying the inequality $\frac{x + 1}{x - 1} \ge \frac{x + 5}{x + 1}$ is __________.

$$x \in (-\infty, -1) \cup (1, 3]$$