Solve for x x^3-3x>0

Problem

x3−3*x>0

Solution

1. Factor the expression on the left side by taking out the greatest common factor, which is x

x*(x2−3)>0

2. Factor further using the difference of squares pattern, a2−b2=(a−b)*(a+b) where a=x and b=√(,3)

x*(x−√(,3))*(x+√(,3))>0

3. Identify the critical points by setting each factor equal to zero.

x=0

x=√(,3)

x=−√(,3)

4. Test intervals created by the critical points (−∞,−√(,3)) (−√(,3),0) (0,√(,3)) and (√(,3),∞) to see where the product is positive.

• For x∈(−∞,−√(,3)) pick x=−2 (−2)*(−2−√(,3))*(−2+√(,3))=(−)*(−)*(−)=negative

• For x∈(−√(,3),0) pick x=−1 (−1)*(−1−√(,3))*(−1+√(,3))=(−)*(−)*(+)=positive

• For x∈(0,√(,3)) pick x=1 (1)*(1−√(,3))*(1+√(,3))=(+)*(−)*(+)=negative

• For x∈(√(,3),∞) pick x=2 (2)*(2−√(,3))*(2+√(,3))=(+)*(+)*(+)=positive

5. Select the intervals where the expression is greater than zero.

x∈(−√(,3),0)∪(√(,3),∞)

Final Answer

x∈(−√(,3),0)∪(√(,3),∞)

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