Convert to Interval Notation x^3<4x

Problem

x3<4*x

Solution

1. Rearrange the inequality to one side so that the other side is zero.

x3−4*x<0

2. Factor the expression by first pulling out the greatest common factor, x

x*(x2−4)<0

3. Factor further using the difference of squares formula a2−b2=(a−b)*(a+b)

x*(x−2)*(x+2)<0

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

x=0

x=2

x=−2

5. Test the intervals created by the critical points (−∞,−2) (−2,0) (0,2) and (2,∞) to see where the expression is negative.

• For x=−3 (−3)*(−3−2)*(−3+2)=(−3)*(−5)*(−1)=−15 (Negative)

• For x=−1 (−1)*(−1−2)*(−1+2)=(−1)*(−3)*(1)=3 (Positive)

• For x=1 (1)*(1−2)*(1+2)=(1)*(−1)*(3)=−3 (Negative)

• For x=3 (3)*(3−2)*(3+2)=(3)*(1)*(5)=15 (Positive)

6. Select the intervals that satisfy the inequality x3−4*x<0

(−∞,−2)∪(0,2)

Final Answer

x3<4*x⇒(−∞,−2)∪(0,2)

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