Page 98 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I inequAlities

3.33 |x − 4| < 1 holds whenever |x − 2| <a holds, find the range of the positive
2
number a .

Solution: Let A = {x : |x − 2| < a, a > 0},B = {x : |x − 4| < 1},
2
√ √
√
then A = {x :2 − a< x< 2+ a, a > 0},B = {x : − 5 <x< − 3, 3 < x< √ 5} .
Since A ⊆ B, we have
√
2 − a> − 5
√
2+ a< − 3
or
√
2 − a> 3
√
2+ a< 5
⇔

√
a< 2+ 5
√
a< −2 − 3
or

√
a< 2 − 3
√
a< 5 − 2
√
which implies 0 <a < 5 − 2 since a> 0.

√
3.34 Solve the inequality x − 3x +2 >x − 3.
2

Solution: The inequality is equivalent to

x − 3 < 0
2
x − 3x +2 ≥ 0
or

x − 3 ≥ 0
2
x − 3x +2 > (x − 3) 2
⇒

x< 3
x ≤ 1 or x ≥ 2
or

x ≥ 3
x> 7/3
⇒ x ≤ 1 or 2 ≤ x< 3 or x ≥ 3.

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