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19. \( 15+|2 x+3| \leq 24 \)

Ask by Best Salazar. in Mexico
Feb 19,2025

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La solución de la desigualdad \( 15 + |2x + 3| \leq 24 \) es \( x \) entre -6 y 3, incluyendo ambos extremos.

Solución

Para resolver la desigualdad \( 15 + |2x + 3| \leq 24 \), seguimos estos pasos: 1. **Restar 15 de ambos lados**: \[ |2x + 3| \leq 24 - 15 \] Esto simplifica a: \[ |2x + 3| \leq 9 \] 2. **Resolver la desigualdad de valor absoluto**: La desigualdad \( |A| \leq B \) se traduce en dos desigualdades: \[ -B \leq A \leq B \] En nuestro caso, esto se convierte en: \[ -9 \leq 2x + 3 \leq 9 \] 3. **Descomponer en dos desigualdades**: - Para la parte izquierda: \[ -9 \leq 2x + 3 \] Restamos 3 de ambos lados: \[ -9 - 3 \leq 2x \] \[ -12 \leq 2x \] Dividimos entre 2: \[ -6 \leq x \quad \text{o} \quad x \geq -6 \] - Para la parte derecha: \[ 2x + 3 \leq 9 \] Restamos 3 de ambos lados: \[ 2x \leq 9 - 3 \] \[ 2x \leq 6 \] Dividimos entre 2: \[ x \leq 3 \] 4. **Combinar las soluciones**: Juntando ambas partes, tenemos: \[ -6 \leq x \leq 3 \] Por lo tanto, la solución de la desigualdad \( 15 + |2x + 3| \leq 24 \) es: \[ x \in [-6, 3] \]

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Bonus Knowledge

To solve \( 15 + |2x + 3| \leq 24 \), first, subtract 15 from both sides to isolate the absolute value: \[ |2x + 3| \leq 9. \] This leads to two possible cases: 1. \( 2x + 3 \leq 9 \) 2. \( 2x + 3 \geq -9 \) Solving these, we get: 1. From \( 2x + 3 \leq 9 \) we find \( 2x \leq 6 \) or \( x \leq 3 \). 2. From \( 2x + 3 \geq -9 \) we find \( 2x \geq -12 \) or \( x \geq -6 \). Thus, the solution to the inequality is \( -6 \leq x \leq 3 \). To visualize this, you can imagine a number line with two important points, -6 and 3, where you would shade the section between them, including the endpoints, indicating all values of \( x \) that satisfy your inequality. Such tools can help solidify the understanding that absolute values can create these "gaps" where we need to consider both positive and negative outcomes.

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