Solve the non-linear inequality:
x^3-5x^2-14x<0
x^3-5x^2-14x<0
x(x^2 - 5x - 14) < 0
x(x-7)(x+2) < 0
so we have 3 critical values:
x = 0, -2, and 7
investigate each domain
1. x < -2 , I will pick x = -5
(-)(-)(-) which is < 0 , that works
2. between -2 and 0, I will pick x = -1
(-)(-1)(+) > 0 , no good
3. between 0 and 7, I will pick x = 1
(+)(-)(+) < 0 , yup, that works
4. x > 7, I will pick x = 10
(+)(+)(+) > 0 , no good
so x < -2 OR 0 < x < 7
notice I did not actually have to find the algebraic answer, just determine if the result is + or -
thanks!
To solve the inequality, x^3 - 5x^2 - 14x < 0, we first need to find the critical points or values of x that divide the number line into regions. In each of these regions, we will determine if the polynomial is positive or negative.
Step 1: Factor the expression if possible.
Unfortunately, x^3 - 5x^2 - 14x does not have a simple factorization.
Step 2: Solve for the critical values.
To find the critical values of x, we set the expression equal to zero and solve for x.
x^3 - 5x^2 - 14x = 0
We can factor out an x from the equation:
x(x^2 - 5x - 14) = 0
Using the quadratic formula or factoring, we find the solutions are x = 0, x = 7, and x = -2.
Step 3: Determine the sign of the polynomial in each region.
We now have four regions on the number line to consider: x < -2, -2 < x < 0, 0 < x < 7, and x > 7.
For x < -2:
Choose any value less than -2, for example, x = -3.
Evaluate the polynomial expression: (-3)^3 - 5(-3)^2 - 14(-3)
We get -9 - 45 + 42 = -12, which is negative.
For -2 < x < 0:
Choose any value between -2 and 0, for example, x = -1.
Evaluate the polynomial expression: (-1)^3 - 5(-1)^2 - 14(-1)
We get -1 - 5 + 14 = 8, which is positive.
For 0 < x < 7:
Choose any value between 0 and 7, for example, x = 2.
Evaluate the polynomial expression: (2)^3 - 5(2)^2 - 14(2)
We get 8 - 20 - 28 = -40, which is negative.
For x > 7:
Choose any value greater than 7, for example, x = 8.
Evaluate the polynomial expression: (8)^3 - 5(8)^2 - 14(8)
We get 512 - 320 - 112 = 80, which is positive.
Step 4: Determine the solution.
From the sign analysis, we can see that the polynomial is negative for x < -2 and 0 < x < 7.
Therefore, the solution to the inequality x^3 - 5x^2 - 14x < 0 is -2 < x < 0 or x > 7.