Find the range of values of x for which x4 + x < x3 + x

To find the range of values of x for which x^4 + x < x^3 + x, we first simplify the inequality:

x^4 + x < x^3 + x

Subtract x and x^3 from both sides to isolate terms:

x^4 - x^3 < 0

Factor out an x^3:

x^3(x - 1) < 0

Now we need to find the values of x for which this expression is less than zero. The expression equals zero when x = 0 or x = 1.

To find the intervals where the expression is less than zero, we test values in the intervals (-∞, 0), (0, 1), and (1, ∞).

For x in (-∞, 0), we can take x = -1 as a test point. Substituting x = -1 into the expression gives (-1)^3((-1) - 1) = -2, which is less than zero. So the interval (-∞, 0) is part of the solution.

For x in (0, 1), we can take x = 0.5 as a test point. Substituting x = 0.5 into the expression gives (0.5)^3((0.5) - 1) = -0.0625, which is less than zero. So the interval (0, 1) is also part of the solution.

For x in (1, ∞), we can take x = 2 as a test point. Substituting x = 2 into the expression gives (2)^3((2) - 1) = 8, which is greater than zero. So the interval (1, ∞) is not part of the solution.

Therefore, the range of values of x for which x^4 + x < x^3 + x is (-∞, 0) ∪ (0, 1).