Page 91 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I inequAlities

3.7 Solve the inequality lg(x − x − 6) < lg(2 − 3x).
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Solution: The inequality holds if and only if

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x − x − 6 > 0
2 − 3x> 0
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x − x − 6 < 2 − 3x
⇒

x< −2 or x> 3
x< 2
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−4 <x< 2
⇒−4 <x< −2, which is the solution of the original inequality.

3.8 a, b are real numbers and a + b =2, show a + b ≤ 2.
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Proof: Suppose a + b> 2, then b> 2 − a ⇒ b > (2 − a) =8 − 12a +6a − a ⇒ a + b > 8 − 12a +6a =
3
3
2
3
2
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b> 2 − a ⇒ b > (2 − a) =8 − 12a +6a − a ⇒ a + b > 8 − 12a +6a =
2
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6a − 12a +6+2 = 6(a − 1) +2 > 2
b> 2 − a ⇒ b > (2 − a) =8 − 12a +6a − a ⇒ a + b > 8 − 12a +6a = 6a − 12a +6+2 = 6(a − 1) +2 > 2 , a contradiction to
2
3
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2
2
2

2
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6a − 12a +6+2 = 6(a − 1) +2 > 2 a + b =2. Hence, a + b ≤ 2.
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