Para resolver la desigualdad $$ 5x^{3} - 4x - 1 < x - 1 $$, seguimos los siguientes pasos: ### Paso 1: Reorganizar la desigualdad Primero, trasladamos todos los términos al lado izquierdo de la desigualdad: \[ 5x^{3} - 4x - 1 - x + 1 < 0 \] Simplificando los términos semejantes: \[ 5x^{3} - 5x < 0 \] ### Paso 2: Factorizar la expresión Factorizamos el término común: \[ 5x(x^{2} - 1) < 0 \] Reconociendo que $$ x^{2} - 1 $$ es una diferencia de cuadrados, lo factorizamos adicionalmente: \[ 5x(x - 1)(x + 1) < 0 \] ### Paso 3: Identificar los puntos críticos Los puntos críticos donde la expresión cambia de signo son las raíces de la ecuación $$ 5x(x - 1)(x + 1) = 0 $$: \[ x = -1, \quad x = 0, \quad x = 1 \] ### Paso 4: Analizar los intervalos Dividimos la recta numérica en intervalos basados en los puntos críticos y determinamos el signo de la expresión en cada intervalo: 1. **Intervalo $$ x 0 \quad \text{(Falso)} \] 3. **Intervalo $$ 0 1 $$:** - Tomemos $$ x = 2 $$: \[ 5(2)(2 - 1)(2 + 1) = 5(2)(1)(3) = 30 > 0 \quad \text{(Falso)} \] ### Paso 5: Determinar la solución La desigualdad $$ 5x^{3} - 5x < 0 $$ se cumple en los intervalos donde la expresión es negativa: \[ x < -1 \quad \text{o} \quad 0 < x < 1 \] ### Conclusión La solución de la desigualdad $$ 5x^{3} - 4x - 1 < x - 1 $$ es: \[ x \in (-\infty, -1) \cup (0, 1) \] Es decir, todos los valores de $$ x $$ que son menores que $$-1$$ o que están entre $$0$$ y $$1$$ satisfacen la desigualdad. La solución de la desigualdad $$5x^{3} - 4x - 1 < x - 1$$ es $$x < -1$$ o $$0 < x < 1$$.

[Solved]: 3) 5x^3-4x-1<x-1

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La solución de la desigualdad

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The Deep Dive

To solve the inequality

, we can first rearrange it as

. This simplifies to

, which factors into

. Further factoring gives

. This product will be negative when one of the factors is positive and the others are negative.

To analyze the critical points, we set each factor equal to zero:

, and

. Testing intervals around these points helps determine where the inequality holds:

For : all factors are negative → positive product.
For : is negative, and are negative → negative product.
For : is positive, is negative, is positive → negative product.
For : all factors are positive → positive product.

Thus, the solution for

and

So the final solution is

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