Convert to Interval Notation x^3<49x

Problem

x3<49*x

Solution

1. Rearrange the inequality to one side by subtracting 49*x from both sides to set the expression against zero.

x3−49*x<0

2. Factor the expression by taking out the greatest common factor, which is x

x*(x2−49)<0

3. Factor further using the difference of squares formula a2−b2=(a−b)*(a+b) for the term (x2−49)

x*(x−7)*(x+7)<0

4. Identify the critical points where the expression equals zero, which are x=0 x=7 and x=−7

5. Test intervals created by these points ((−∞,−7) (−7,0) (0,7) and (7,∞) to see where the product is negative.

• For x∈(−∞,−7) the product is (−)*(−)*(−)=negative

• For x∈(−7,0) the product is (−)*(−)*(+)=positive

• For x∈(0,7) the product is (+)*(−)*(+)=negative

• For x∈(7,∞) the product is (+)*(+)*(+)=positive

6. Select the intervals that satisfy the inequality <0

(−∞,−7)∪(0,7)

Final Answer

x3<49*x⇒(−∞,−7)∪(0,7)

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