# Equation and Inequation Exercise

Bài tập toán tiếng Anh chương Phương trình và bất phương trình. Những từ vựng quan trọng là: equation, inequation, variable, equal, real, express, expression, sum, product, positive, negative, value, satisfy, possible

1. The middle number of three increasing consecutive odd numbers is $n$. Express the product of the three numbers in terms of $n$.
2. The product of two numbers is 10. One of them is $a$. Express their sum in terms of $a$.
3. The first of three increasing consecutive even numbers is $2n – 4$. Express the last number in terms of $n.$
4. Identify the expressions that are always positive.
1. $a^2.$
2. $a+2.$
3. $|a+1|.$
4. $a^2+1.$
5. $4-(-a)^3.$
5. Non-negative values $a$ and $b$ satisfy $a + b = 0$. Find the values of $a$ and $b.$
6. Assume $|x|=3,|y|=10,$ and $xy$ Find all possible values of $x-y.$
7. Given equation $|x – y| + |y – z| = 0$, identity true statements about the variables.
1. All variables are zero.
2. All variables are equal.
3. Exactly two variables are equal.
4. At least two variables are equal.
8. The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure. Which expression represents the perimeter of the figure?
1. $2x+3yz$.
2. $2x+2y+yz$.
3. $5xyz$.
4. $x^2+y^3z$.
9. At the beginning of her mathematics class, Mrs. Reno gives a warm-up problem. She says, “I am thinking of a number such that $6$ less than the product of $7$ and this number is $85$.” Which number is she thinking of?
10. Robin spent 17USD at an amusement park for admission and rides. If she paid 5USD for admission, and rides cost 3USD each, what is the total number of rides that she went on?
1. 2.
2. 4.
3. 12.
4. 9.
11. A girl can ski down a hill five times as fast as she can climb up the same hill. If she can climb up the hill and ski down in a total of 9 minutes, how many minutes does it take her to climb up the hill?
1. 4.5.
2. 1.8.
3. 7.2.
4. 7.5.
12. Mr. Perez owns a sneaker store. He bought 350 pairs of basketball sneakers and 150 pairs of soccer sneakers from the manufacturers for 62,500USD. He sold all the sneakers and made a 25% profit. If he sold the soccer sneakers for 130USD per pair, how much did he charge for one pair of basketball sneakers?
13. Find all real values of $x$ satisfy $|x|+3-|x+3|=6.$
14. The solutions of equation $33x^2 + 99x – 9999 = 0$ are $x_1$ and $x_2$. Compute the value of $(x_1 – 1)(x_2 – 1)$ without solving the equation.
15. Write a quadratic function with only rational coefcients and a zero of $1+\sqrt{3}.$
16. The two sides of a right triangle are 2 and 3. Find all possible values of the third side.
17. The product of two consecutive odd integers are 143. Find the smaller integer.
18. Find all integral values of $x$ for which $\displaystyle \frac{12(x^2-4x+3)}{x^3-3x^2-x+3}$ has a positive integral value.
19. Find all real values of $x$ that satisfy $\frac{x^3-x^2-x+1}{x^3-x^2+x-1}=0.$
20. Find all positive real values of $x$ satisfy $\frac{1}{x+\sqrt{x}}+\frac{1}{x-\sqrt{x}} \leqslant 1.$ Hướng dẫn. Note that $x \geqslant 0$ for $\sqrt{x}$ to be meaningful and $x\ne 0,1$ for the denominators to be nonzero. Rationalizing denominators and adding the resulting fractions yields $\frac{2x}{x^2-x} \leqslant 1$ or $\frac{2x}{x(x-1)} \leqslant 1.$ Since $x\ne 0,\frac{2}{x-1} \leqslant 1$. If $x>1, 2 \leqslant x-1$ or $x \geqslant 3.$ If $0<x<1, 2 \geqslant x-1$ or $x \leqslant 3$, so $0<x<1.$ Thus the solution set is $\{x\big| 0<x<1 \text{ x } \geqslant 3\}.$
21. Find the minimum value of $1+3(3-x)^2.$
22. Find the minimum value of $y =x+\frac{1}{x}$, where $x>0.$
23. Two positive values $x$ and $y$ satisfy $x + y = 1$. Find the minimum of $\frac{1}{x}+\frac{1}{y}.$
24. Find the minimum values of the expressions $|x-3|+|5-x|.$
25. Values of $a$ and $b$ satisfy $(a+1) ^2+(b-23)^2=0.$ Compute the value of $a^b.$
26. Suppose $x + y = 6$ and $xy = 4$. Find the value of $x^2y + xy^2$ without solving the equations.
27. Suppose $a + b = 1$ and $a^2 + b^2 = 2.$ Find the value of $a^3 + b^3$ without solving the equations.
28. Suppose $a<-2$. Simplify expression $|2-|1-a||.$
29. Suppose $(x – a)(x + 2) = (x + 6)(x – b)$ is true for all $x\in \mathbb{R}.$ Find the values of $a$ and $b$.
30. If we increase $x$ and $y$ by $10\%$, by what percent does $\dfrac{x}{x+y}$ change?
31. Evaluate the value of $\displaystyle \frac{x^2 -y^2 }{x^2y+xy^2}$ at $x=\sqrt{5}+1$ and $y=\sqrt{5}-1.$
32. Find all ordered pairs of integers $(x,y)$ that satisfy $x^2+y^2 \leqslant 25$ and $y=x-3.$
33. A product was discounted twice by the same percentage. The original price was 100 USD and the current price is 81 USD. Find the discount percentage.
34. The sum of two numbers is 21 and their product $-7$. Find (i) the sum of their squares, (ii) the sum of their reciprocals and (iii) the sum of their fourth powers.
35. Find positive integers $a$ and $b$ with $\sqrt{5+\sqrt{24}}=\sqrt{a}+\sqrt{b}$.
36. Given that $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\frac{1}{\sqrt{99}+\sqrt{100}}$ is an integer, find it.
37. Prove that if $a,b,c$ are non-negative real numbers then $(a+b)(b+c)(c+a) \geqslant 8abc$.
38. A train, $x$ meters long, traveling at a constant speed, takes $20$ seconds from the time it first enters a tunnel $300$ meters long until the time it completely emerges from the tunnel. One of the stationary ceiling lights in the tunnel is directly above the train for 10 seconds. Find the value of $x$.
39. If $a,b,$ and $c$ are different numbers and if $a^3+3a+14=0, b^3+3b+14=0$, and $c^3+3c+14=0$, find the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.
40. Some people agree to share in the cost of buying a boat. If ten of them later decide not to buy in, each of those remaining would have to chip in one dollar more. If the sole payment actually occurs after an additional fifteen people drop out, each of those ultimately remaining has to pay two dollars more than he would have had to pay had only the first ten dropped out. How many people originally agreed to buy the boat?
41. Function $f(x) = ax^2 + bx + c$. Its two zeros $r_1$ and $r_2$ satisfy $1 < r_1 < 2 < r_2$. Find the sign of product $f(1)\cdot f(2).$
42. $0<x<y$ are integers such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{2015}$. The number of pairs $(x,y)$ is
1. $12$
2. $13$
3. $14$
4. $11$
43. If the quadratic equation $x^2+ax+6a=0$ has only integer roots, then the number of values of $a$ is
1. $7$
2. $8$
3. $9$
4. $10$
44. If the system of linear equations $x + y + z = 6, x + 2y + 3z = 14$ and $2x + 5y +\lambda z = \mu$, ($\lambda, \mu \in \mathbb{R}$) has no solution, then
1. $\lambda \ne 8$
2. $\lambda = 8,\mu \ne 36$
3. $\lambda = 8,\mu =36$
4. None of these
45. Consider the equation $x^4 – 18x^3 + kx^2 + 174x- 2015 = 0$. If the product of two of the four roots of the equation is $-31$, find the value of $k$.