3x to the third power<24x
If this is
3x3 < 24x
then divide by 3x on each side
x2 < 8
x < sqrt8
To solve the inequality 3x^3 < 24x, we first want to simplify the equation. Dividing both sides by 3x gives:
3x^3 / 3x < 24x / 3x
Simplifying further, we get:
x^2 < 8x
Now, to solve for x, we want to isolate x on one side of the inequality. Subtracting 8x from both sides gives:
x^2 - 8x < 0
Now, we have a quadratic inequality. We want to find the values of x that make the expression less than 0. One way to solve this is by factoring:
x(x - 8) < 0
Now, we have two factors x and (x - 8). We can set up a number line and analyze the sign of each factor:
On the number line, we mark two critical points: 0 and 8.
For the factor x, if x < 0, then x is negative. If x > 0, then x is positive.
For the factor (x - 8), if x < 8, then (x - 8) is negative. If x > 8, then (x - 8) is positive.
Now, let's consider the different combinations:
1. x < 0 and (x - 8) < 0:
In this case, both x and (x - 8) are negative. Here, our inequality x(x - 8) < 0 is satisfied since a negative number multiplied by another negative number is positive.
2. x > 8 and (x - 8) > 0:
In this case, both x and (x - 8) are positive. Here, our inequality x(x - 8) < 0 is not satisfied since a positive number multiplied by another positive number is positive.
Therefore, the solution to the inequality 3x^3 < 24x is x < 0.