-3x^3+3x<0 — step-by-step solution | Explain MAths

-3x^3+3x<0

1) Let's solve in

ℝ

:

- 3 x^{3} + 3 x < 0

You can write the expression:

3 x - 3 x^{3}

in factored form:

x \cdot (3 - 3 x^{2})

We study the sign of each factor of the product

⇥The expression

x

is zero at

x = 0

, it is negative before and positive after

Let's study the sign of

3 - 3 x^{2}

by widening the sign to ≤ and solving in

ℝ

:

3 - 3 x^{2} \leq 0

Since

-3 < 0

, divide both sides by

-3

and reverse the inequality:

- 1 + x^{2} \geq 0

An even power only depends on the absolute value:

x^{2} \geq 1

is equivalent to

| x | \geq 1

| A | \geq B

means that:

A \geq B

or

A \leq - B

2) Let's draw up the sign table:

x	$-∞$		$-1$		$0$		$1$		$+∞$
$x$		-		-	0	+		+
$3 - 3 x^{2}$		-	0	+		+	0	-
$- 3 x^{3} + 3 x$		+	0	-	0	+	0	-

▶Solution found:

x \in (-1, 0) \cup (1, +∞)