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**Unformatted text preview: **for extraneous solutions. Sure enough, we
see that x = 1 does not satisfy the original equation and must be discarded. Our solutions
are x = − 1 and x = 0.
2
2. To solve the inequality, it may be tempting to begin as we did with the equation − namely
by multiplying both sides by the quantity (x − 1). The problem is that, depending on x,
(x − 1) may be positive (which doesn’t aﬀect the inequality) or (x − 1) could be negative
(which would reverse the inequality). Instead of working by cases, we collect all of the terms
on one side of the inequality with 0 on the other and make a sign diagram using the technique
given on page 247 in Section 4.2. 268 Rational Functions x3 − 2x + 1
x−1
3 − 2x + 1
x
1
− x+1
x−1
2
3 − 2x + 1 − x(x − 1) + 1(2(x − 1))
2x
2(x − 1)
2x3 − x2 − x
2x − 2 ≥ 1
x−1
2 ≥0
≥0 get a common denominator ≥0 expand Viewing the left hand side as a rational function r(x) we make a sign diagram. The only
value excluded from the domain of r is x = 1 which is the sol...

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