Convert to Interval Notation 1+2/(x+1)<=2/x
Problem
Solution
Move all terms to one side to set the inequality against zero.
Find a common denominator, which is
x*(x+1) to combine the terms into a single fraction.
Simplify the numerator by expanding and combining like terms.
Factor the numerator to identify the critical points of the expression.
Identify the critical points where the numerator is zero (
x=−2,1 and where the denominator is zero (x=−1,0 . Note that the expression is undefined atx=−1 andx=0 Test the intervals created by the critical points
(−∞,−2] [−2,−1) (−1,0) (0,1] and[1,∞) to see where the inequality holds true.
For
x=−3 ((−1)*(−4))/((−3)*(−2))=4/6>0 (False)For
x=−1.5 ((0.5)*(−2.5))/((−1.5)*(−0.5))=(−1.25)/0.75<0 (True)For
x=−0.5 ((1.5)*(−1.5))/((−0.5)*(0.5))=(−2.25)/(−0.25)>0 (False)For
x=0.5 ((2.5)*(−0.5))/((0.5)*(1.5))=(−1.25)/0.75<0 (True)For
x=2 ((4)*(1))/((2)*(3))=4/6>0 (False)
Combine the intervals that satisfy the inequality, using brackets for included points (where the numerator is zero) and parentheses for excluded points (where the denominator is zero).
Final Answer
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