Convert to Interval Notation x^4>100x^2

Problem

x4>100*x2

Solution

1. Rearrange the inequality to one side by subtracting 100*x2 from both sides to set the expression against zero.

x4−100*x2>0

2. Factor out the greatest common factor, which is x2

x2*(x2−100)>0

3. Factor further using the difference of squares formula a2−b2=(a−b)*(a+b) for the term (x2−100)

x2*(x−10)*(x+10)>0

4. Identify the critical points where the expression equals zero, which are x=0 x=10 and x=−10

5. Test intervals created by these points ((−∞,−10) (−10,0) (0,10) and (10,∞) to see where the inequality holds true.

• For (−∞,−10) let x=−11 (−11)2*(−11−10)*(−11+10)=121*(−21)*(−1)>0 (True)

• For (−10,0) let x=−1 (−1)2*(−1−10)*(−1+10)=1*(−11)*(9)<0 (False)

• For (0,10) let x=1 (1)2*(1−10)*(1+10)=1*(−9)*(11)<0 (False)

• For (10,∞) let x=11 (11)2*(11−10)*(11+10)=121*(1)*(21)>0 (True)

6. Combine the intervals where the expression is strictly greater than zero.

Final Answer

x4>100*x2⇒(−∞,−10)∪(10,∞)

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