Unformatted text preview: ution to 2x − 2 = 0. The zeros
of r are the solutions to 2x3 − x2 − x = 0, which we have already found to be x = 0, x = − 1
2
and x = 1, the latter was discounted as a zero because it is not in the domain. Choosing test
values in each test interval, we construct the sign diagram below.
(+) 0 (−) 0 (+)
−1
2 0 (+)
1 1
We are interested in where r(x) ≥ 0. We ﬁnd r(x) > 0, or (+), on the intervals −∞, − 2 ,
(0, 1) and (1, ∞). We add to these intervals the zeros of r, − 1 and 0, to get our ﬁnal solution:
2
−∞, − 1 ∪ [0, 1) ∪ (1, ∞).
2
3 2
1
3. Geometrically, if we set f (x) = x −−x+1 and g (x) = 2 x − 1, the solutions to f (x) = g (x) are
x1
the xcoordinates of the points where the graphs of y = f (x) and y = g (x) intersect. The
solution to f (x) ≥ g (x) represents not only where the graphs meet, but the intervals over
which the graph of y = f (x) is above (>) the graph of g (x). We obtain the graphs below. The ‘Intersect’ command conﬁrms that the graphs cross when x = − 1...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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