Solve |x 1| >2| x^3|

Evaluate

∣ x \times 1 ∣ > 2 x^{3}

Any expression multiplied by 1 remains the same

∣ x ∣ > 2 x^{3}

Move the expression to the left side

∣ x ∣ - 2 x^{3} > 0

Separate the inequality into 4 possible cases

x - 2 x^{3} > 0, x \geq 0, x^{3} \geq 0 x - 2 (- x^{3}) > 0, x \geq 0, x^{3} < 0 - x - 2 x^{3} > 0, x < 0, x^{3} \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

Evaluate

More Steps

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x^{3} \geq 0 x - 2 (- x^{3}) > 0, x \geq 0, x^{3} < 0 - x - 2 x^{3} > 0, x < 0, x^{3} \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

The only way a base raised to an odd power can be greater than or equal to 0 is if the base is greater than or equal to 0

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x - 2 (- x^{3}) > 0, x \geq 0, x^{3} < 0 - x - 2 x^{3} > 0, x < 0, x^{3} \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

Evaluate

More Steps

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x > 0, x \geq 0, x^{3} < 0 - x - 2 x^{3} > 0, x < 0, x^{3} \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

The only way a base raised to an odd power can be less than 0 is if the base is less than 0

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x > 0, x \geq 0, x < 0 - x - 2 x^{3} > 0, x < 0, x^{3} \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

Evaluate

More Steps

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x > 0, x \geq 0, x < 0 x < 0, x < 0, x^{3} \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

The only way a base raised to an odd power can be greater than or equal to 0 is if the base is greater than or equal to 0

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x > 0, x \geq 0, x < 0 x < 0, x < 0, x \geq 0 - x - 2 (- x^{3}) > 0, x < 0, x^{3} < 0

Evaluate

More Steps

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x > 0, x \geq 0, x < 0 x < 0, x < 0, x \geq 0 x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty), x < 0, x^{3} < 0

The only way a base raised to an odd power can be less than 0 is if the base is less than 0

x \in (- \infty, - \frac{2}{2}) \cup (0, \frac{2}{2}), x \geq 0, x \geq 0 x > 0, x \geq 0, x < 0 x < 0, x < 0, x \geq 0 x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty), x < 0, x < 0

Find the intersection

0 < x < \frac{2}{2} x > 0, x \geq 0, x < 0 x < 0, x < 0, x \geq 0 x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty), x < 0, x < 0

Find the intersection

0 < x < \frac{2}{2} x \in \emptyset x < 0, x < 0, x \geq 0 x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty), x < 0, x < 0

Find the intersection

0 < x < \frac{2}{2} x \in \emptyset x \in \emptyset x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty), x < 0, x < 0

Find the intersection

0 < x < \frac{2}{2} x \in \emptyset x \in \emptyset - \frac{2}{2} < x < 0

Solution

x \in (- \frac{2}{2}, 0) \cup (0, \frac{2}{2})

Solve the inequality