solve the the following inequality. write your solution in interval notation.
x^3-49x<0
_ (I don't know how to make the sign on my computer)
x^3 - 49x < 0
x(x^2 - 49) < 0
x(x+7)(x-7) < 0
consider f(x) = x(x+7)(x-7)
the x-intercepts would be -7, 0, 7
Knowing the basic shape of a cubic, we have
x < -7 OR 0 < x < 7
I will let you write that in the interval notation that you learned.
To solve the inequality x^3 - 49x < 0, we can start by factoring out an x from both terms:
x(x^2 - 49) < 0
Now, we need to determine the values of x that satisfy this inequality. Since the product of two factors is negative when one factor is negative and the other is positive, we have two cases to consider.
Case 1: x > 0
In this case, x is positive. Then, both x and (x^2 - 49) are positive, since subtracting a positive number from x^2 will not change its sign.
Case 2: x < 0
In this case, x is negative. Then, x is negative, but (x^2 - 49) is positive. We can determine the values of x by solving the equation (x^2 - 49) = 0:
x^2 - 49 = 0
(x - 7)(x + 7) = 0
So, the solutions for this case are x = -7 and x = 7.
Now, let's combine the solutions from both cases:
Case 1: x > 0,
Case 2: x < 0
The solutions in interval notation are:
(-∞, -7) U (0, 7)