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in equalities

by satish_iitg » Thu Aug 08, 2013 2:04 pm
Which of the following inequalities has a solution that when graphed on the number line, is a line segment of finite length ?

a x^4 >= 1
b x^3 <= 27
c x^2 >= 16
d 2 <= |x| <= 5
e 2 <= 3x+4 <= 6[/img]
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by Matt@VeritasPrep » Thu Aug 08, 2013 3:56 pm
The basic idea here is we want a single inequality with a floor and a ceiling: say 3 > x > 1, or -100 < x < 100. Anything that only has a floor (say x > 3) will extend to infinity.

A:: x� ≥ 1 implies x ≥ 1 or -1 ≥ x, so there are two lines, each of infinite length (1 to infinity, and -1 to -infinity)

B:: 27 ≥ x³ implies 3 ≥ x, so we have a line of infinite length (3 to -infinity)

C:: x² ≥ 16 implies x ≥4 or -4 ≥ x ... same as A, pretty much

D:: 5 ≥ |x| ≥ 2 implies 5 ≥ x ≥ 2 or -2 ≥ x ≥ -5, so we have finite line segments, but there are TWO of them.

E:: 6 ≥ 3x + 4 ≥ 2 implies 2/3 ≥ x ≥ -2/3 ... success! One line segment, of length 4/3 (from 2/3 to -2/3).
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