Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
[tex]x^2+8 x+15\ \textgreater \ 0[/tex]
Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Answer (1)
Factor the quadratic expression: x 2 + 8 x + 15 = ( x + 3 ) ( x + 5 ) .
Find the roots of the equation ( x + 3 ) ( x + 5 ) = 0 , which are x = − 5 and x = − 3 .
Test the intervals ( − ∞ , − 5 ) , ( − 5 , − 3 ) , and ( − 3 , ∞ ) to determine where the inequality 0"> ( x + 3 ) ( x + 5 ) > 0 holds.
Express the solution set in interval notation: ( − ∞ , − 5 ) ∪ ( − 3 , ∞ ) .
( − ∞ , − 5 ) ∪ ( − 3 , ∞ )
Explanation
Analyze the problem We are given the polynomial inequality 0"> x 2 + 8 x + 15 > 0 . Our goal is to find the solution set for x and express it in interval notation. First, we need to factor the quadratic expression.
Factor the quadratic To factor the quadratic expression x 2 + 8 x + 15 , we look for two numbers that multiply to 15 and add to 8. These numbers are 3 and 5. Therefore, we can factor the expression as ( x + 3 ) ( x + 5 ) .
Find the roots Now we have the inequality 0"> ( x + 3 ) ( x + 5 ) > 0 . To find the intervals where this inequality holds, we need to find the roots of the quadratic equation ( x + 3 ) ( x + 5 ) = 0 . The roots are x = − 3 and x = − 5 .
Test the intervals We now test the intervals determined by the roots x = − 5 and x = − 3 . The intervals are ( − ∞ , − 5 ) , ( − 5 , − 3 ) , and ( − 3 , ∞ ) .
For the interval ( − ∞ , − 5 ) , let's test x = − 6 . Then, 0"> ( x + 3 ) ( x + 5 ) = ( − 6 + 3 ) ( − 6 + 5 ) = ( − 3 ) ( − 1 ) = 3 > 0 . So, the inequality holds in this interval.
For the interval ( − 5 , − 3 ) , let's test x = − 4 . Then, ( x + 3 ) ( x + 5 ) = ( − 4 + 3 ) ( − 4 + 5 ) = ( − 1 ) ( 1 ) = − 1 < 0 . So, the inequality does not hold in this interval.
For the interval ( − 3 , ∞ ) , let's test x = 0 . Then, 0"> ( x + 3 ) ( x + 5 ) = ( 0 + 3 ) ( 0 + 5 ) = ( 3 ) ( 5 ) = 15 > 0 . So, the inequality holds in this interval.
Express the solution in interval notation The inequality 0"> x 2 + 8 x + 15 > 0 holds for the intervals ( − ∞ , − 5 ) and ( − 3 , ∞ ) . Therefore, the solution set in interval notation is ( − ∞ , − 5 ) ∪ ( − 3 , ∞ ) .
Final Answer The solution set for the inequality 0"> x 2 + 8 x + 15 > 0 is ( − ∞ , − 5 ) ∪ ( − 3 , ∞ ) .
Examples
Understanding polynomial inequalities is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers need to ensure that the structure can withstand certain loads. This often involves solving inequalities to determine the range of loads the bridge can safely handle. Similarly, in physics, analyzing the motion of objects under certain conditions may require solving polynomial inequalities to determine when certain conditions are met, such as the object's height exceeding a certain threshold.
