# How do you solve the inequality #-x < 2x^3 < -8x^3#?

There are no solutions to the given inequality

so there can be no solutions to these inequalities

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To solve the inequality -x < 2x^3 < -8x^3, we need to consider each part separately.

First, let's focus on the inequality -x < 2x^3:

-x < 2x^3 0 < 2x^3 + x 0 < x(2x^2 + 1)

This inequality holds true when x > 0 because both x and the expression 2x^2 + 1 are positive.

Next, let's examine the inequality 2x^3 < -8x^3:

2x^3 < -8x^3 0 < -8x^3 - 2x^3 0 < -10x^3

This inequality holds true when x < 0 because -10x^3 is negative when x is negative.

Combining both conditions, we find that x must satisfy both x > 0 and x < 0 simultaneously. However, there are no real numbers that satisfy both conditions, so there are no solutions to the given inequality.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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