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Answer:
[tex](-\infty, -7)\cup(-2, \infty)[/tex]
Step-by-step explanation:
Inequalities are mathematical statements that use greater than or equal to, less than or equal to, greater than, less than, or not equal to symbols instead of the equal symbol.
1. Greater than. [tex]>[/tex]
2. Less than. [tex]<[/tex]
3. Greater than or equal to. [tex]\geq[/tex]
4. Less than or equal to. [tex]\leq[/tex]
5. Not equal to. [tex]\ne[/tex]
Solving inequalities is somehow similar to solving equations however the final answer is a set of numbers.
1. Change the inequality symbol to an equal symbol.
2. Solve for the critical values.
3. Pick a representative from each interval and substitute the representative into the function, then solve.
4. Form the set notation for the interval that satisfies the original function.
First
[tex]\begin{aligned}x^2+9x+14=0\end{aligned}[/tex]
Second
[tex]\begin{aligned}x^2+9x+14&=0\\(x+7)(x+2)&=0\\x+7&=0\\x&=-7\\&\text{or}\\x+2&=0\\x&=-2\end{aligned}[/tex]
The critical values are -7 and -2.
Third
Refer to the Figure Below. The representatives that we can pick from each interval are -8, -5, and 0.
[tex]\begin{aligned}(-8)^2+9(-8)+14&\stackrel{?}{>}0\\64-72+14&\stackrel{?}{>}0\\6&>0\:\text{true}\end{aligned}[/tex]
[tex]\begin{aligned}(-5)^2+9(-5)+14&\stackrel{?}{>}0\\25-45+14&\stackrel{?}{>}0\\-6&>0\:\text{False}\end{aligned}[/tex]
[tex]\begin{aligned}(0)^2+9(0)+14&\stackrel{?}{>}0\\0+0+14&\stackrel{?}{>}0\\14&>0\:\text{true}\end{aligned}[/tex]
Fourth
This means to say that the intervals that satisfy the function are [tex](-\infty, -7)[/tex] and [tex](-2, \infty)[/tex]. We are using open parenthesis since the inequality symbol is >. To form the final answer, take the union of the two intervals.
[tex](-\infty, -7)\cup(-2, \infty)[/tex]
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