Problem Book in High-School
Mathematics
Edited by A.I. PRILEPKO, D.Sc.
MIR Publishers Moscow
Translated from the Russian by I.A. Aleksanova
First published 1985
Second printing 1989
Reviewed from the 1982 Russian edition
Printed in the Union of Soviet Socialist Republics
Preface
The present problem book is meant for high-school students who intend to enter techical colleges. It contains more than two thousand problems and examples covering all divisions of high-school mathematics.
The main aim of the book is to help students to revise their school knowledge of mathematics and develop a technique in solving a variety of proplems.
Chapter 1 : RATIONAL EQUATIONS, INEQUALITIES AND FUNCTIONS IN ONE VARIABLE
1.1. Linear Equations and Inequalities in One Variable.
A Linear Function
Solve the following equations:
2. \(\frac{x}{2}+\frac{x}{6}+\frac{x}{12}+\frac{x}{20}+\frac{x}{30}+\frac{x}{42}=-6.\)
3. (a) \(3x+1=(4x-3)-(x-4);\)
\(x \in \mathbb{R}\) (Any real number)
3. (b) \(3x+1=(4x-3)-(x-5).\)
\(\emptyset\) (No solution)
4. \(ax=a^2.\)
\(a \text{ for } a \ne 0; \quad x \in \mathbb{R} \text{ for } a=0\)
5. \((a-2)x=a^2-4.\)
\(a+2\;for\;a \ne 2;\;x \in \mathbb{R}\; for\;a=2\)
6. \((a^2-9)x=a^3+27.\)
\(\frac{a^2-3a+9}{a-3}\;for\;a \in (-\infty;-3)\cup(-3;3)\cup(3;\infty);\;x \in \mathbb{R}\; for\;a=-3;\;\emptyset\; for\;a = 3.\)
Solve the following inequalities:
7. (a) \(7x>3;\)
\((\frac{3}{7};\infty);\)
7. (b) \(-4x>5;\)
\((-\infty;-\frac{5}{4});\)
7. (c) \(5x+6 \le 3x-8;\)
7. (d) \(\frac{x}{2}+1 \le \frac{x}{\sqrt{3}}+\frac{1}{2}.\)
\([3+2\sqrt{3};\infty).\)
8. (a) \(ax \le 1;\)
\([\frac{1}{a};\infty) for\;a \in (-\infty;0), \mathbb{R}\;for\;a = 0, (-\infty; \frac{1}{a}]\;for\;a \in (0;\infty);\)
8. (b) \(ax \gt 1.\)
\((-\infty;\frac{1}{a})\;for\;a \in (-\infty;0),\;\emptyset\;for\;a=0,\;(\frac{1}{a};\infty)\;for\;a \in (0;\infty).\)
Solve the following systems of inequalities:
9. (a) \(\begin{cases} 3x > 1, \\ -x < 3;\end{cases}\)
\((\frac{1}{3};\infty);\)
9. (b) \(\begin{cases} 2x < \pi, \\ -x > -1.6;\end{cases}\)
\((-\infty;\frac{\pi}{2});\)