Convert to Interval Notation (x+2)/(x+3)<(x-1)/(x-2)

Problem

(x+2)/(x+3)<(x−1)/(x−2)

Solution

1. Move all terms to one side to set the inequality against zero.

(x+2)/(x+3)−(x−1)/(x−2)<0

2. Find a common denominator to combine the fractions.

((x+2)*(x−2)−(x−1)*(x+3))/((x+3)*(x−2))<0

3. Expand the numerators using the FOIL method.

((x2−4)−(x2+2*x−3))/((x+3)*(x−2))<0

4. Simplify the numerator by distributing the negative sign and combining like terms.

(x2−4−x2−2*x+3)/((x+3)*(x−2))<0

(−2*x−1)/((x+3)*(x−2))<0

5. Identify the critical points where the numerator or denominator equals zero.

−2*x−1=0⇒x=−1/2

x+3=0⇒x=−3

x−2=0⇒x=2

6. Test the intervals created by the critical points (−3,−1/2,2) in the inequality.
   For x<−3 test x=−4 (−2*(−4)−1)/((−4+3)*(−4−2))=7/((−1)*(−6))=7/6>0 (False)
   For −3<x<−1/2 test x=−1 (−2*(−1)−1)/((−1+3)*(−1−2))=1/((2)*(−3))=−1/6<0 (True)
   For −1/2<x<2 test x=0 (−2*(0)−1)/((0+3)*(0−2))=(−1)/((3)*(−2))=1/6>0 (False)
   For x>2 test x=3 (−2*(3)−1)/((3+3)*(3−2))=(−7)/((6)*(1))=−7/6<0 (True)

7. Determine the solution set based on the intervals that satisfy the inequality.
   The inequality is satisfied on (−3,−1/2)∪(2,∞)

Final Answer

(x+2)/(x+3)<(x−1)/(x−2)⇒(−3,−1/2)∪(2,∞)

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