Question

How do you solve `x^4<4x^2`?

Answer 1

`x in (-2, 0) uu (0, 2)`

i.e. `-2 < x < 0` or `0 < x < 2`

Explanation:

First note that if `x = 0` then `x^4 = 0 = 4x^2`, so the inequality is false.

If `x != 0` then `x^2 > 0` and we can divide both sides of the inequality by `x^2` to get:

`x^2 < 4`

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Case `bb(x > 0)`

Since `y = sqrt(x)` is a strictly monotonically increasing function for `x > 0`, we can take the square root of both sides of our simplified inequality to find:

`x < 2`

Hence we have solutions:

`0 < x < 2`

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Case `bb(x < 0)`

Note that `x^2 = (-x)^2` and `-x > 0`, so we find:

`-x < 2`

Multiplying both sides by `-1` and reversing the inequality, we get:

`x > -2`

Hence we have solutions:

`-2 < x < 0`

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Background

The truth or falsity of an inequality is unaltered by any of the following operations:

• Add or subtract the same value from both sides.
• Multiply or divide both sides by the same positive value.
• Multiply or divide both sides by the same negative value and reverse the inequality (`<` becomes `>`, `>=` becomes `<=`, etc.).

More generally:

• Apply the same strictly monotonically increasing function to both sides of the inequality.
• Apply the same strictly monotonically decreasing function to both sides of the inequality and reverse the inequality.