solve the inequality express in interval notation
x^4 + 4x^3 - 9x^2 - 36x < 0
x(x^3 + 4x^2 - 9x - 36) < 0
x( x^2(x+4) - 9(x+4) ) = 0
x(x+4)(x^2-9) < 0
x(x+4)(x+3)(x-3) < 0
so your critical values are
x = 0, -3, -4, and 3
Knowing that y = x^4 + .... is a quartic that rises in the 1st and 2nd quadrant because of the +1x^4 term, and it has 4 x-intercepts, you can make a rough sketch showing a W shape with those intercepts.
It should be obvious that the graph is below the x-axis ( y < 0 ) for values between -3 and -4 or between 0 and 3
so in standard notation:
-4 < x < -3 OR 0 < x < 3
I will leave it up to you to express it in interval notation
Thank you!
To solve the inequality x^4 + 4x^3 - 9x^2 - 36x < 0, we need to determine the intervals of x values that make the inequality true.
Step 1: Factor the expression on the left-hand side to simplify the inequality:
x^4 + 4x^3 - 9x^2 - 36x = x(x^3 + 4x^2 - 9x - 36)
Step 2: Additional factoring of the expression can be done:
x(x^3 + 4x^2 - 9x - 36) = x(x^2(x + 4) - 9(x + 4))
Now, we have x(x^2 - 9)(x + 4) = x(x - 3)(x + 3)(x + 4)
Step 3: Find the critical points by setting each factor equal to zero:
x = 0, x = -3, x = 3, and x = -4
Step 4: Plot these critical points on the number line:
-4 --(-3)---0---(3)---
Step 5: Test each interval created by the critical points:
For the interval (-∞, -4): Pick a value x = -5, and substitute it into the inequality:
(-5)(-5 - 3)(-5 + 3)(-5 + 4) = -5(8)(-2)(-1) = 80
Since the expression is positive, it does not satisfy the inequality.
For the interval (-4, -3): Pick a value x = -3.5, and substitute it into the inequality:
(-3.5)(-3.5 - 3)(-3.5 + 3)(-3.5 + 4) = -3.5(-6.5)(-0.5)(0.5) = 4.53125
Since the expression is positive, it does not satisfy the inequality.
For the interval (-3, 0): Pick a value x = -2, and substitute it into the inequality:
(-2)(-2 - 3)(-2 + 3)(-2 + 4) = -2(-5)(1)(2) = 20
Since the expression is negative, it satisfies the inequality.
For the interval (0, 3): Pick a value x = 1, and substitute it into the inequality:
(1)(1 - 3)(1 + 3)(1 + 4) = 1(-2)(4)(5) = -40
Since the expression is negative, it satisfies the inequality.
For the interval (3, ∞): Pick a value x = 4, and substitute it into the inequality:
(4)(4 - 3)(4 + 3)(4 + 4) = 4(1)(7)(8) = 224
Since the expression is positive, it does not satisfy the inequality.
Step 6: Write the final solution in interval notation:
The solution to the inequality x^4 + 4x^3 - 9x^2 - 36x < 0 is (-3, 0) U (3, ∞).