Solve the nonlinear inequality. Express the solution using interval notation. [tex]x^2\ \textgreater \ 2(x+4)[/tex]

Rewrite the inequality: 0"> x 2 − 2 x − 8 > 0 .
Factor the quadratic: 0"> ( x − 4 ) ( x + 2 ) > 0 .
Find critical points: x = − 2 , 4 .
Express the solution in interval notation: ( − ∞ , − 2 ) ∪ ( 4 , ∞ ) .

Explanation

Problem Setup We are given the nonlinear inequality 2(x+4)"> x 2 > 2 ( x + 4 ) . Our goal is to solve for x and express the solution in interval notation.

Rewriting the Inequality First, we rewrite the inequality to have 0 on one side. We distribute the 2 on the right side to get 2x + 8"> x 2 > 2 x + 8 . Then, we subtract 2 x and 8 from both sides to obtain 0"> x 2 − 2 x − 8 > 0 .

Factoring the Quadratic Next, we factor the quadratic expression x 2 − 2 x − 8 . We are looking for two numbers that multiply to -8 and add to -2. These numbers are -4 and 2. Thus, we can factor the quadratic as 0"> ( x − 4 ) ( x + 2 ) > 0 .

Finding Critical Points and Intervals Now, we find the critical points by setting each factor equal to zero: x − 4 = 0 and x + 2 = 0 . Solving these equations gives us x = 4 and x = − 2 . These critical points divide the number line into three intervals: ( − ∞ , − 2 ) , ( − 2 , 4 ) , and ( 4 , ∞ ) .

Testing Intervals We now test a value from each interval in the inequality 0"> ( x − 4 ) ( x + 2 ) > 0 to determine where the inequality is true.

For the interval ( − ∞ , − 2 ) , let's test x = − 3 : 0"> ( − 3 − 4 ) ( − 3 + 2 ) = ( − 7 ) ( − 1 ) = 7 > 0 . So the inequality is true on this interval.

For the interval ( − 2 , 4 ) , let's test x = 0 : ( 0 − 4 ) ( 0 + 2 ) = ( − 4 ) ( 2 ) = − 8 < 0 . So the inequality is false on this interval.

For the interval ( 4 , ∞ ) , let's test x = 5 : 0"> ( 5 − 4 ) ( 5 + 2 ) = ( 1 ) ( 7 ) = 7 > 0 . So the inequality is true on this interval.

Expressing the Solution Since the inequality is true on the intervals ( − ∞ , − 2 ) and ( 4 , ∞ ) , the solution in interval notation is ( − ∞ , − 2 ) ∪ ( 4 , ∞ ) .


Examples
Understanding inequalities is crucial in many real-world applications. For instance, consider a scenario where a company's profit is modeled by a quadratic function, and they want to determine the range of production levels that will ensure a profit above a certain threshold. Solving a quadratic inequality, similar to the one in this problem, would help them find the production quantities that satisfy the desired profit condition. This ensures the company operates within profitable parameters, optimizing their business strategy.

Answered by GinnyAnswer | 2025-07-03

To solve the inequality \ 2(x + 4)"> x 2 > 2 ( x + 4 ) , first rewrite it as \ 0"> x 2 − 2 x − 8 > 0 and factor it to \ 0"> ( x − 4 ) ( x + 2 ) > 0 . The critical points are x = − 2 and x = 4 , which divide the number line into intervals. Testing the intervals, the solution is ( − ∞ , − 2 ) ∪ ( 4 , ∞ ) .
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Answered by Anonymous | 2025-07-04