Math topic中文

Rational inequalities

Solving inequalities of the form $\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)} > 0$ where $\mathrm{f}(x)$ and $\mathrm{g}(x)$ are linear expressions or quadratic expressions.

Practice questions

1The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
2The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
3Find the exact solution of the inequality $\frac{2x+1}{x^2-1} \le 1.$ A curve $G$ has equation $ay^2 + bx^3 + cx = 2$ , w…
4The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
5The function $g$ is defined by $g : x \mapsto \ln(x - \sqrt{3})$ , for $x \in \mathbb{R}$ , $x > \sqrt{3}$ . Explain how yo…
6The curve C is given by the equation $\dfrac{y^2}{x^2}=\dfrac{3a-x}{a+x},\quad x \ne 0,$ where $a$ is a positive const…
7已知集合 $A=\{x\in z||x|\leqslant 2\}$ ， $B=\left\{ x\left|\dfrac{x-2}{x+2}\leqslant 0\right.\right\}$ ，则 $(\quad)$ A. $A∩B=A$ …
8Sketch the graph of $y = \frac{7-2x}{3x^2 + x - 14}$ . Give the equations of the asymptotes and the coordinates of the tu…
9Without using a calculator, solve the inequality $\frac{3x^2 + 9x - 8}{x - 1} < 2.$ Hence solve the following inequali…
10Show algebraically that $x^2 + 2x + 2$ is always positive for all real values of $x$ . Without using a calculator, solve…
11集合 $P=\{x|\dfrac{x}{x-3}< 0\}$ ， $M=\{x|x^{2}< 9\}$ ，则 $P∩M=$ ___.
12A curve $C$ has equation $y = \frac{1}{\sqrt{bx^2 - 2bx}}$ , where $b < 0$ . Sketch $C$ and give, in terms of $b$ where ap…
13Do not use a calculator in answering this question. Solve the inequality $\frac{6x^2 + 2x - 3}{2x - 1} \ge 2(x + 1).$ …
14Sketch the curve with equation $y = \ln(x+1), x > -1$ , stating the equation of asymptote and intercepts with the axes. O…
15The function $\text{f}$ is defined by $\text{f} : x \mapsto \frac{1}{x^2 - x - 6} + 2, \quad x \in \mathbb{R}, \ x \ne -2, \ x \ne 3.$ …
16Do not use a calculator to solve this question. Solve the inequality $\frac{x-6}{4x^2+x-5} \ge 1$ . Hence solve the i…
17Find the set of values of $x$ such that $-1<\dfrac{2x+3}{x-1}<1$ . Deduce the range of values of $x$ such that $-1<\dfrac{2x+1}{x-2}<1$ …
18Without using a calculator, solve $\frac{30-11x}{x^2-9} \le -2.$ Solve $(a-3bx^2)e^{ax-bx^3} < 0,$ where $a$ and $b$ …
19It is given that the curve $C$ has equation $y = \dfrac{-4x^2 - 6x - 11}{2x + 2}$ , $x \in \mathbb{R}$ , $x \ne -1$ . Witho…
20MI Promo 9758/2020/PU1/Q2 (modified (ii)) Without the use of a calculator, solve the inequality $\frac{3}{x - 1} \ge 2x + 3.$ …
21The function $\text{g}$ is defined by $\text{g} : x \mapsto \ln(x - \sqrt{3})$ , for $x \in \mathbb{R}$ , $x > \sqrt{3}$ .…
22On the same axes, sketch the graphs of $y = \frac{1}{x - 2}$ and $y = 4x - 2$ , indicating clearly the equations of the a…
23By using the substitution $y = |x|$ , solve the inequality $\frac{|x|+3}{x^2+1} > 1.$
24Sketch the curve $C$ with equation $y = \frac{x-2}{x-1}$ , stating the exact coordinates of any points of intersection wi…
25The diagram shows the graphs of $y = |a - bx|$ and $y = |mx - 6|$ where $a$ , $b$ and $m$ are positive constants. The gra…
26已知集合 $M=\{x|\dfrac{2}{x-1}\geqslant 0\}.$ 集合 $N=\{x|2x+3>0\}$ ，则 $(∁_{R}M)∩N=(\quad)$ A. $[-\dfrac{3}{2},1)$ B. $(-\dfrac{3}{2},1)$ …
27MI Promo 9758/2020/PU1/Q2 (modified (ii)) Without the use of a calculator, solve the inequality $\dfrac{3}{x - 1} \ge 2x + 3$ …
28It is given that $\mathrm{f}(x) = \ln(1 + 2x^2)$ . A polynomial $p(x) = ax^2 + bx + c$ is used to approximate $\mathrm{f}(x)$ …
29下面命题错误的是 A. 命题 $p\vee q$ 为假命题，则 $p$ ， $q$ 均为假命题 B. 已知 $p:\dfrac{1}{x} < 1$ ， $q:-1\leqslant x\leqslant 1$ ，则 $\neg p$ 是 $q$ …
30Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
31Given that $a \in \mathbb{R}$ and $a > 1$ , use an algebraic method to solve the inequality $\frac{ax - a + 2}{(x - a)(1 - x)} \le 1 .$ …
32Given that both $a$ and $b$ are positive constants, on the same diagram, sketch the graphs of $y = \frac{2}{|x - a|} \quad \text{and} \quad y = b|x - a|.$ …
33The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto 1 - \frac{4}{x^2}, \text{ for } x \in \mathbb{R}, x < -1.$ …
34Two fair n-sided dice, each with faces numbered 1, 2, ..., n, (where $n \ge 4$ ) are tossed in an experiment. The n…
35A family of curves is given by the differential equation $x\dfrac{dy}{dx}-y=x(x-y)$ . By substituting $y=vx$ , show that t…
36By expressing $\frac{3x - 7}{(x + 3)(x - 2)} - 1$ as a single simplified fraction, solve exactly the inequality $\frac{3x - 7}{(x + 3)(x - 2)} < 1.$ …
37Without using a calculator, solve the inequality $\dfrac{4x}{x-3} \ge 1. Hence find the range of values of$ x $that satisfies$ …
38The curve $C$ has equation $y = \frac{2x^2 - 10x + 17}{x - 3}, \quad x \in \mathbb{R}, \quad x \ne 3$ . Without using a c…
39Without using a calculator, solve the inequality $\dfrac{11x-19}{x^2+2x-8} \le 2$ . Hence solve $\dfrac{11e^{-x}-19}{e^{-2x}+2e^{-x}-8} \le 2$ …
40Solve the inequality $\dfrac{x-5}{x-1} \le -1$ . Hence solve $\dfrac{\cos (x-4)}{\cos x} \le -1$ for $0 \le x \le \pi$ .
41设数列 $\{a_{n}\}$ 满足 $a_{n+1}= \dfrac {4a_{n}-2}{a_{n}+1}(n∈N^{*})(1)$ 若 $a_{1}=3$ ， $b_{n}= \dfrac {2-a_{n}}{a_{n}-1}(n∈N^{*})$ …
42In this question, $k$ is a positive constant. Using an algebraic method, solve the inequality $\frac{2x^2 - 2k^2}{x} < x + k$ …
43Without using a calculator, solve $1 + \frac{6}{x} \ge x$ , $1 + \frac{6}{|x|} \ge |x|$ .
44不等式 $\dfrac {3x+1}{1-2x}\geqslant 0$ 的解集是 ___ ．
45The function g is defined by $g : x \mapsto \ln\bigl(x - \sqrt{3}\bigr)$ , for $x \in \mathbb{R}$ , $x > \sqrt{3}$ . Explai…
46An arithmetic series has first term $a$ and common difference $d$ , where $a > 0$ and $d \ne 0$ . The first, sixth and nin…
47Solve the inequality $\displaystyle \frac{x^{2} - ax - a}{x - a} \ge a$ , where $a$ is a positive real constant, leaving…
48Find the set of values of $\theta$ lying in the interval $-\dfrac{\pi}{2}\!<\!\theta\!<\!\dfrac{\pi}{2}$ such that the s…
49Without the use of a calculator, solve the inequality $\frac{(x^2 - 2x + 4)(x - 3)}{(x + 2)} \ge 0.$ Hence solve the i…
50Without the use of a calculator, solve the inequality $\frac{3x^2 + 1}{2x^2 + 3x - 2} > 1$ . Hence solve the inequality $\frac{3x^2 + 1}{2x^2 + 3|x| - 2} > 1$ …
51Sketch the graphs of $y = 1 + \frac{a-2}{x-a}$ and $y = -\frac{1}{a}x + \frac{2}{a}$ on a single diagram, where $a$ is a…
52By using the substitution $y = |x|$ , solve the inequality $\frac{|x| + 3}{x^2 + 1} > 1$ .
53The curve $C$ has equation $y=\frac{ax^2-2ax+a-2}{x-2},\quad a\in\mathbb{R},\ a\ne 0.$ Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ …
54Without using a calculator, solve the inequality $\frac{2x^2 + 3x}{2x^2 + x - 1} \le \frac{1}{2x + 2}$ . Using your answe…
55The function f is defined as follows. f : $x -> \displaystyle -\frac{1}{x^{2}+1}$ for $x \in \mathbb{R}$ . Sketch the gra…
56The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{aligned} \mathrm{f} : x &\mapsto \frac{x^2 + 2x}{2x^2 - 5x + 2}, \quad x \in \mathbb{R}^+, \ x \ne \frac{1}{2}, 2, \\ \mathrm{g} : x &\mapsto \frac{-3}{2x^2 - 5x + 2}, \quad x \in \mathbb{R}^+, \ x \ne \frac{1}{2}, 2. \end{aligned}$ …
57If $x^3 + 5x^2 + 8x + 4 = (x + 2)^2(x + 1)$ , find the range of values of $x$ for which $\frac{18}{1-x} \ge x^2 + 6x + 14$ …
58Express $\dfrac{33-8x}{x^2+2x-15}+2$ as a single algebraic fraction. Hence, without using a calculator, solve exactly th…
59The diagram shows the graphs of $y = |a - b x|$ and $y = |m x - 6|$ where $a, b$ and $m$ are positive constants. The gra…
60A geometric series has common ratio $r$ , and an arithmetic series has first term $a$ and common difference $d$ , where $a$ …