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Solve the inequality x(x + 6) > 16 - x + 14 + x^2. Write your answer in interval notation.

 Dec 18, 2023
 #1
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We can solve the inequality x(x + 6) > 16 - x + 14 + x^2 by following these steps:

 

Simplify the right side:

 

Combining constant terms on the right:

 

x(x + 6) > 16 + x^2

 

Expand the left side:

 

x * (x + 6) = x^2 + 6x

 

Move all terms to one side:

 

x^2 + 6x - (x^2 + 6x) > 16

0 > 16 (contradiction)

 

 

Analyze the result:

 

Since we obtain a contradiction (always false statement) after moving all terms to one side, there are no real values of x that satisfy the inequality.

 

Write the solution in interval notation:

 

Therefore, the solution is the empty set:

 

x ∈ ∅

 

This means there are no intervals which represent the solution, as there are no valid real numbers satisfying the inequality.

 Dec 18, 2023

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