Inequalities

Exercises

1. Find the real numbers which satisfy the following inequalities (or equalities): a) $5 - x^2 < 8$ b) $5 - x^2 < -2$ c) $(x - 1)(x - 3) > 0$ d) $\frac{1}{x} + \frac{1}{1-x} > 0$ e) $\frac{x-1}{x+1} > 0$ f) $x^3 > x$ g) $-5 \leq 3 - 2x \leq 9$ h) $2x - 3 < x + 4 < 3x - 2$

2. Find the real numbers which satisfy the following systems of inequalities (or equalities): a) $\begin{cases} -(x^2 + x) \geq 0 \ 3x + x^2 \geq 0 \end{cases}$ [Answer: $x = 0$]

   b) $\begin{cases} x^2(2x - 1) > 0 \ \frac{3x + 2}{x^2 + 4} \leq 0 \end{cases}$ [Answer: No solutions]

   c) $\begin{cases} \frac{2}{x} + \frac{1}{x^2} > -1 \ \frac{x^2 + x + 4}{4 - x^2} \geq 0 \end{cases}$ [Answer: $-2 < x < 2$]

3. Find the real numbers which satisfy the following inequalities (or equalities): a) $|x - 3| = 8$ b) $|x - 3| < 8$ c) $|x + 4| < 2$ d) $|x - 1| + |x + 1| < 2$ e) $|x - 1| + |x - 2| > 1$ f) $|x - 1| - |x + 2| = 3$ g) $|x - 1||x + 1| = 0$ h) $\left|3x + \frac{x}{x-1} - 1\right| \geq 1$ i) $\left|\frac{x^2-9}{x^2-4x+3}\right| \geq 0$

4. Find the real numbers which satisfy the following inequalities: a) $\sqrt{x^2 - 4} \leq 2 - x$ b) $\sqrt{x^2 - 2x + 1} > 2 - |x + 4|$ c) $\sqrt{x^2 - x + 1} > |x + 3|$ d) $\sqrt{x + 1} + \sqrt{x - 1} < 1$ e) $x - 2 \leq \sqrt{\frac{x^2-1}{x+2}}$ f) $\sqrt[3]{x(x^2 - 1)} > x - 1$

5. Prove that for every fixed number $a \geq -1$, the inequality $(1+a)^n \geq 1+na$ holds for every $n \in \mathbb{N}$. (Bernoulli’s Inequality)