What is the solution to the inequality [tex]x^2\ \textgreater \ -9 x-18[/tex]?
A) [tex]x\ \textless \ -6[/tex] and [tex]x\ \textgreater \ -3[/tex]
B) [tex]-6\ \textless \ x\ \textless \ -3[/tex]
C) [tex]x \leq-6[/tex] and [tex]x \geq-3[/tex]
D) [tex]x\ \textless \ -6[/tex] or [tex]x\ \textgreater \ -3[/tex]
Answer (2)
Rewrite the inequality: 0"> x 2 + 9 x + 18 > 0 .
Factor the quadratic: 0"> ( x + 6 ) ( x + 3 ) > 0 .
Find the roots: x = − 6 and x = − 3 .
Determine the intervals where the inequality holds: x < − 6 or -3"> x > − 3 . The final answer is -3}"> x < − 6 or x > − 3 .
Explanation
Rewrite the Inequality We are given the inequality -9x - 18"> x 2 > − 9 x − 18 . Our goal is to find the values of x that satisfy this inequality. First, we rewrite the inequality so that one side is zero. This gives us 0"> x 2 + 9 x + 18 > 0 .
Factor the Quadratic Next, we factor the quadratic expression x 2 + 9 x + 18 . We are looking for two numbers that multiply to 18 and add to 9. These numbers are 6 and 3. So, we can factor the expression as 0"> ( x + 6 ) ( x + 3 ) > 0 .
Find the Critical Points Now, we need to find the intervals where the expression ( x + 6 ) ( x + 3 ) is positive. The roots of the quadratic equation ( x + 6 ) ( x + 3 ) = 0 are x = − 6 and x = − 3 . These roots divide the number line into three intervals: x < − 6 , − 6 < x < − 3 , and -3"> x > − 3 .
Test Intervals We test a value in each interval to see if the inequality 0"> ( x + 6 ) ( x + 3 ) > 0 is satisfied.
If x < − 6 , let's test x = − 7 . Then 0"> ( x + 6 ) ( x + 3 ) = ( − 7 + 6 ) ( − 7 + 3 ) = ( − 1 ) ( − 4 ) = 4 > 0 . So, the inequality is satisfied for x < − 6 .
If − 6 < x < − 3 , let's test x = − 4 . Then ( x + 6 ) ( x + 3 ) = ( − 4 + 6 ) ( − 4 + 3 ) = ( 2 ) ( − 1 ) = − 2 < 0 . So, the inequality is not satisfied for − 6 < x < − 3 .
If -3"> x > − 3 , let's test x = 0 . Then 0"> ( x + 6 ) ( x + 3 ) = ( 0 + 6 ) ( 0 + 3 ) = ( 6 ) ( 3 ) = 18 > 0 . So, the inequality is satisfied for -3"> x > − 3 .
State the Solution Therefore, the solution to the inequality -9x - 18"> x 2 > − 9 x − 18 is x < − 6 or -3"> x > − 3 .
Examples
Understanding quadratic inequalities helps in various real-world scenarios. For instance, if a company's profit is modeled by a quadratic function, solving a quadratic inequality can determine the range of sales needed to achieve a certain profit level. Similarly, in physics, analyzing the trajectory of a projectile often involves solving quadratic inequalities to find when the projectile reaches a certain height or distance. These applications highlight the practical importance of mastering quadratic inequalities.
The solution to the inequality -9x - 18"> x 2 > − 9 x − 18 is given by -3"> x < − 6 or x > − 3 . This results from factoring the quadratic and testing intervals. The choice that matches this solution is option D.
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