Page 92 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I inequAlities

3.9 The rational numbers a, b, c, d satisfy d >c (i), a + b = c + d (ii), a + d< b + c (iii),
determine the order of these four numbers.

Solution: (i) ⇒ b + d >b + c . This together with (iii) implies a + d< b + d , thus a<b .
(iii)-(ii) ⇒ d − b< b − d ⇒ d< b . (ii) ⇒ b − d = c − a , since b − d> 0, then c − a> 0,
i.e. c>a . As a conclusion, we obtain the order a<c < d<b .

2
3.10 If the inequality ax + bx − 6 < 0 has the solution set {x|− 2 < x< 3}, find the
values of a and b .

2
Solution: The condition implies that the equation ax + bx − 6 =0 has two roots
x = −2,x =3. Vieta’s formulas imply
a
−2 +3 = −
b
6
(−2) × 3= −
a
from which we can obtain a =1,b = −1.

3.11 Given 2x +6y ≤ 15,x ≥ 0,y ≥ 0, find the maximum value of 4x +3y .

15−2x 5 1 15 15
Solution: 2x +6y ≤ 15 ⇔ y ≤ = − x , thus 4x +3y ≤ 4x + − x =3x + .
6 2 3 2 2
5 1 15 15 15
y ≥ 0 ⇒ − x ≥ 0 ⇒ x ≤ . Hence, 4x +3y ≤ 3 × + = 30, which implies that
2 3 2 2 2
the maximum value of 4x +3y is 30.

3.12 Given (m + 1)x − 2(m − 1)x + 3(m − 1) < 0, find all real values of m such that
2
the inequality has no solution.

Solution: The inequality has no solution if and only if

2
Δ=4(m − 1) − 12(m + 1)(m − 1) ≤ 0
m +1 > 0

⇒

2
m + m − 2 ≥ 0
m +1 > 0
⇒

m ≤−2 or m ≥ 1
m> −1

⇒ m ≥ 1.

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