How do you solve and write the following in interval notation: #x^2+4x12>0#?
graph{x^2+4x12 [26.85, 24.5, 17.23, 8.45]}
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To solve the inequality (x^2 + 4x  12 > 0), first find the critical points by setting the expression equal to zero and solving for (x). Then, use these critical points to create intervals and test points within each interval to determine where the inequality holds true. Finally, write the solution in interval notation.

Find critical points: Set (x^2 + 4x  12 = 0) and solve for (x).
(x^2 + 4x  12 = 0) factors as ((x + 6)(x  2) = 0).
The critical points are (x = 6) and (x = 2).

Create intervals: Use the critical points to divide the number line into intervals: (( \infty, 6)), ((6, 2)), and ((2, +\infty)).

Test each interval: Choose a test point within each interval and evaluate (x^2 + 4x  12) to determine if it's greater than zero.
 Test (x = 7) in the interval (( \infty, 6)): ((7)^2 + 4(7)  12 = 49  28  12 = 9 > 0).
 Test (x = 0) in the interval ((6, 2)): (0^2 + 4(0)  12 = 12 < 0).
 Test (x = 3) in the interval ((2, +\infty)): (3^2 + 4(3)  12 = 9 + 12  12 = 9 > 0).

Write solution in interval notation: The inequality holds true when (x) is in the intervals where (x^2 + 4x  12 > 0), which are (( \infty, 6)) and ((2, +\infty)).
So, in interval notation, the solution is: (( \infty, 6) \cup (2, +\infty)).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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