不等式(英文)

Linear Inequalities in One Unknown

1. Meanings of the inequality signs ‘’ and ‘’

(a) The inequality sign ‘’ means ‘is greater than or equal to’

e.g.

a5 means a is greater than or equal to 5.

(b) The inequality sign ‘’ means ‘is less than or equal to’

e.g.

a3 means a is less than or equal to–3.

2. Solutions of inequalities and their representation on the number line

(a) For an inequality in one unknown x, the values of x that can satisfy the inequality are called the

solutions of the inequality.

(b) The solutions of the linear inequalities in one unknown can be represented on the number line.

e.g. (i) The solutions of

xk can be represented as



k

(ii)The solutions of

xk can be represented as

•

k

(iii) The solutions of

xk can be represented as



k

(iv) The solutions of

xk can be represented as

•

k

3. Basic properties of inequalities

(a) transitive property

If

ab and

bc, then

ac.

e.g.

32 and

21, then

31

(b) additive property

If

ab then

acbc.

e.g.

32 and

3121

(c) multiplication property

(i) If

ab and

c0, then

acbc.

 64 e.g.

32 and

20, then

326 , 224

(ii) If

ab and

c0, then

acbc.

 64 e.g.

32 and

20, then

3(2)6 , 2(2)4

(d) reciprocal property

If

ab0, then

1111. e.g.

320 then

.

ab32(e) the above properties also hold when the inequality signs ‘’ and ‘’ are replaced by ‘’ and ‘’

respectively.

4. Linear inequalities in one unknown and their applications

(a) An inequality which has only one unknown with index 1 is called a linear inequality in one unknown.

(b) We can find the solutions of inequalities systematically by applying the basic properties of

inequalities.

(c) There are many daily life problems that involve the concept of inequalities. Sometimes we can set up

simple inequalities in one unknown to find the relevant solutions, but we must consider if the answers

obtained suit the real situation.

Exercise A

Level 1

1. Fill in each of the following blanks with an inequality sign 「>」、「<」、「」or「」

(a) If x > 4 and 4 > y , then x ____ y.

(b) If 5 < x and x < y, then y ____ 5.

(c) If a > 3 and 3  b , then a ____ b. (d) If a  1 amd 1 < b, then a ____ b.

(e) If a  5 and 5 < x , then a ____ x. (f) If x < 3 then x  1 ____ 4.

(g) If y   2, then y  3 ____ 1.

(h) If x  5, then 5x  2 ____ 23.

(i) If a < 9, then 4a  3 ____ 33.

2. Write down an inequality in x corresponding to each of the following diagrams.

(a) (b)

•



0

5

1

0

____________

(c)



0

8

____________

(d)

___________



0

4

___________

3. Represent the solutions of each of the following inequalities graphically on the number line..

1(a)

x

2

(b)

2x

4. Solve the following linear inequalities in one unknown and represent their solutions graphically on the

number line.

(a) 5  2x > 0

(c) 3x  1 < 2x  5

Level II

(b) 2 < 5  x

(b) x  6 < 2

5. If x > y > 0 , fill in each of the following blanks with an inequality sign ‘>’ or ‘<’.

(a)

11 ________

2y2x (b)

11 _________

3y3x6. If x > y，x > 0 and y < 0, fill in each of the following blanks with an inequality sign.

(a)

113 ______

3

yx (b)

552 _________

2

yx

7. Solve the following linear inequalities in one unknown and represent their solutions graphically on the

number line.

(a) 4x  21 < 3

(c)

1

(e)

(b)

3x14

42x7

3 (d)

3xx15

22x312

5 (f)

5(x7)10

33

(g)

8. Find the largest integer that can satisfy the inequality

9. Find the range of values of x which satisfy both 3x4 > 2(x1) and x < 6.

10. Solve the following inequalities and represent the solutions graphically.

4x159(a)

 (b) –17  5x + 3 < 18

75x3

(c)

3

1xx1

34 (h)

x212x5

436x23x1x.

236x324

4 (d)

x3x42x – 1 <

 – 1

235

11. Find the two smallest consecutive integers whose sum is greater than 35.

12. Peter’s present age is greater than two times his age fifteen years ago. What is his greatest possible

present age?

13. In a Mathematics course, a student will get a pass certificate if his average score in the four tests is at

least 50. If John’s scores in the first three tests are 43, 52 and 48, what is the minimum score he must get in

the fourth test in order to get a pass certificate?

14. There are altogether 15 coins in Mr. Wan’s purse. Each coin is either a $1 coin or a $2 coin. If the total

value of these coins is less than $25, at most how many $2 coins are there in Mr. Wan’s purse?

15. It is known that the relationship between the cost of a book($C) and the total number of pages(P) is

C

1P23. A publisher is going to publish a reference book with the total number of pages not less than

8

200. Estimate the minimum cost of this book.

Level III (Optional)

1. Solve

2x11 for each of the following cases:

x1(a)

x1

(b)

x1

2. Given that

ab

and

b , c0, prove that

aacbbc.