solve the inequality express in interval notation
x^4 + 4x^3 - 9x^2 - 36x < 0
Assistance needed.
Ah yeah which is why I post the question.
Alissa -- Writeacher corrected your spelling of "Algebra" so that a math expert would see your post.
Jiskha doesn't have any experts in Alegra (whatever that is).
x^4 + 4x^3 - 9x^2 - 36x
= x(x^3 + 4x^2 - 9x - 36)
Cubics are hard to solve, so look for easy roots. They will be factors of 36.
A little synthetic division reveals
(x+4)(x+3)x(x-3)
Now since everything is single roots, the graph alternates between positive and negative between roots.
Since it's a quartic with a positive coefficient, it will be positive before the lowest root and after the highest root.
So,
y > 0 for x < -4
y < 0 for -4 < x < -3
y > 0 for -3 < x < 0
y < 0 for 0 < x < 3
y > 0 for x > 3
Thank you I appreciate your help and the fact that you didn't make a big issue out of a typo. Again thank you.
To solve the inequality x^4 + 4x^3 - 9x^2 - 36x < 0 in interval notation, you will first need to find the critical points, which are the values of x where the left side of the inequality is equal to zero.
Step 1: Factor the expression x^4 + 4x^3 - 9x^2 - 36x < 0 by factoring out an x:
x(x^3 + 4x^2 - 9x - 36) < 0
Step 2: Now, let's analyze the sign of the inequality for different intervals.
Interval 1: x < 0
Substitute a value less than 0 into the expression x(x^3 + 4x^2 - 9x - 36) to determine the sign:
(-1)(-1 + 4 + 9 + 36) < 0
(-1)(48) < 0
-48 < 0
Since the expression evaluates to a negative value in Interval 1, it satisfies the inequality.
Interval 2: 0 < x < 3
Substitute a value between 0 and 3 into the expression x(x^3 + 4x^2 - 9x - 36) to determine the sign:
(1)(1 + 4 - 9 - 36) < 0
(1)(-40) < 0
-40 < 0
Since the expression evaluates to a negative value in Interval 2, it satisfies the inequality.
Interval 3: x > 3
Substitute a value greater than 3 into the expression x(x^3 + 4x^2 - 9x - 36) to determine the sign:
(4)(64 + 4(16) - 9(4) - 36) < 0
(4)(64 + 64 - 36 - 36) < 0
(4)(56) < 0
224 < 0
Since the expression evaluates to a positive value in Interval 3, it does not satisfy the inequality.
Step 3: Combine the intervals where the inequality is satisfied.
The solution in interval notation is:
(-∞, 0) ∪ (0, 3)