不等式(英文)

Chapter 1 Linear Inequalities in One Unknown
Name:_________( ) Class: ( )
Revision Notes
1. Meanings of the inequality signs ‘≥’ and ‘≤’
(a) The inequality sign ‘≥’ means ‘is greater than or equal to ’ e.g. 5≥a means a is greater than or equal to 5.
(b) The inequality sign ‘≤’ means ‘is less than or equal to ’ e.g. 3-≤a 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 k x > can be represented as
(ii)The solutions of k x ≥ can be represented as (iii) The solutions of k x < can be represented as
(iv) The solutions of k x ≤ can be represented as
3. Basic properties of inequalities (a) transitive property If b a > and c b >
, then c a >.
e.g. 23> and 1
2>, then 13>
(b) additive property
If b a > then c b c a +>+.
e.g. 23> and 1213+>+
(c) multiplication property (i) If b a > and 0>c , then bc ac >.
e.g. 23> and 02>, then 46 422 , 623>=>=⨯=⨯
(ii) If b a > and 0<c , then bc ac <.
e.g. 23> and 02<-, then 46 4)2(2 , 6)2(3-<-=>-=-⨯-=-⨯
(d) reciprocal property
If 0>>b a , then
b a 11<. e.g. 023>> then 2
131<.
(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)
____________
___________ (c) (d)
____________ ___________
3. Represent the solutions of each of the following inequalities graphically on the number line..
(a) 21
->x
(b) x ≥2
4. Solve the following linear inequalities in one unknown and represent their solutions graphically on the number line. (a) 5 + 2x > 0 (b) x - 6 < -2
(c) 3x - 1 < 2x + 5
(b) 2 < 5 - x
Level II
5. If x > y > 0 , fill in each of the following blanks with an inequality sign ‘>’ or ‘<’. (a)
x
21
________ y 21
(b)
x
31
- _________ y 31-
6. If x > y ，x > 0 and y < 0, fill in each of the following blanks with an inequality sign.
(a)
31
+x
______ 31+y
(b)
25
-x
_________ 25-y
7. Solve the following linear inequalities in one unknown and represent their solutions graphically on the number line. (a) -4x - 21 < 3 (b)
414
3
≥+x (c) 73
21≤-x
(d)
2
5123x x ->+ (e) 215
3
<-+x (f)
3
10
3)7(5-<+-x (g) 4
1
31-+>
+-x x (h)
6
5
23142-≤
--x x
8. Find the largest integer that can satisfy the inequality x x x >+++-3
1
322.
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.
(a) ⎩⎨⎧-≥-<-3579154x x (b) –17 ≤ 5x + 3 < 18
(c) 424
3
63<---<-x (d)
2x – 1 < 343-x ≤5
2x – 1
11. Find the two smallest consecutive integers whose sum is greater than 35.
12. Peter’s present age is greater tha n 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
238
1
+=
P C . A publisher is going to publish a reference book with the total number of pages not less than 200. Estimate the minimum cost of this book.
*********************************************************************************** Level III (Optional) 1.
Solve
11
1
2->++x x for each of the following cases: (a) 1->x (b) 1-<x
(Ans: (a)3
2
->x
(b)1-<x )
2. Given that 0 and >> b , c b a , prove that
c
b c
a b a ++>
. (Hint: To prove A >B , you may prove A –B > 0. ) 3.
Note that 02≥a for any real number a . If a and b are two real numbers, prove that
(a) ab b a 222≥+; (Hint: Using 0)(2
≥-b a ) (b) ab b
a ≥+2
.
4. If a and b are two real numbers, prove that if ab > 0, then
2≥+a b
b a . Hence find the least value of c
b
a b a c a c b ++
+++.
(Ans. (b) 6)
5. A property investment company plans to build residential flats of a total floor area of 24 000 m 2. There are two types of flats, A and B. The areas of type A and type B are 60 m 2 and 80 m 2 respectively. It is
restricted that the number of flats of type A must be at least twice that of type B and there are at least 200 flats of type A. Let x be the number of flats of type A and y be the number of flats of type B. Write down all the constraints on x and y.
(Ans: x y x y x y x , : non-negative integers 608024000
2200+≤≥≥⎧⎨
⎪
⎪⎩
⎪
⎪)。