Let's solve each inequality step-by-step.

First Inequality

$-3 \leq 2x + 5 < 7$

To solve this compound inequality, we break it down into two parts:

1. First part: $-3 \leq 2x + 5$

   • Subtract 5 from both sides: $-3 - 5 \leq 2x$ $-8 \leq 2x$
   • Divide by 2: $-4 \leq x$ or $x \geq -4$

2. Second part: $2x + 5 < 7$

   • Subtract 5 from both sides: $2x < 2$
   • Divide by 2: $x < 1$

Combining both parts, we get: $-4 \leq x < 1$ So, the solution for the first inequality is: $x \in [-4, 1)$

Second Inequality

$2 < 3x - 4 \leq 19$

Similarly, we split this compound inequality into two parts:

1. First part: $2 < 3x - 4$

   • Add 4 to both sides: $6 < 3x$
   • Divide by 3: $2 < x$ or $x > 2$

2. Second part: $3x - 4 \leq 19$

   • Add 4 to both sides: $3x \leq 23$
   • Divide by 3: $x \leq \frac{23}{3}$ which is approximately $x \leq 7.67$.

Combining both parts, we get: $2 < x \leq \frac{23}{3}$ So, the solution for the second inequality is: $x \in (2, \frac{23}{3}]$

If you need further clarification or more details on any step, feel free to ask.

Related Questions

1. How do you solve compound inequalities in general?
2. What is the difference between strict inequalities and non-strict inequalities?
3. How do you represent solutions on a number line?
4. How can compound inequalities be applied in real-life scenarios?
5. What are the steps to combine solutions of inequalities?

Tip

When working with compound inequalities, it’s helpful to split the inequality into two parts and solve each part separately before combining the results.