https://r2.erweima.ai/mathbot/image/1726959489029uc4j1r4e.jpg

The problem in the image asks to solve and isolate \( z \) in the compound inequality:

\[
-7z - 50 \leq -1 \quad \text{and} \quad -6z + 70 > -2
\]

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

### 1. First Inequality: \( -7z - 50 \leq -1 \)

1. Add 50 to both sides:
   \[
   -7z \leq -1 + 50
   \]
   \[
   -7z \leq 49
   \]

2. Divide by -7 (and reverse the inequality sign because we're dividing by a negative number):
   \[
   z \geq \frac{49}{-7}
   \]
   \[
   z \geq -7
   \]

### 2. Second Inequality: \( -6z + 70 > -2 \)

1. Subtract 70 from both sides:
   \[
   -6z > -2 - 70
   \]
   \[
   -6z > -72
   \]

2. Divide by -6 (again, reverse the inequality):
   \[
   z < \frac{-72}{-6}
   \]
   \[
   z < 12
   \]

### Combined Solution

The solution to the system is the intersection of both inequalities:
\[
-7 \leq z < 12
\]

Thus, the correct answer is \( \boxed{B} \).

---

Would you like more details, or do you have any questions? 

Here are five related questions you can explore:
1. How would the solution change if the second inequality was \( \leq \) instead of \( > \)?
2. Can compound inequalities always be represented as intervals?
3. How does reversing the inequality sign work when dividing by a negative number?
4. What happens if both inequalities are strict (i.e., no \( \leq \))?
5. How would the solution change if one of the inequalities had no solution?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Despeja z en la desigualdad compuesta: -7z - 50 ≤ -1 y -6z + 70 > -2.

Learn how to solve the compound inequality -7z - 50 ≤ -1 and -6z + 70 > -2. Follow a detailed, step-by-step explanation, covering key concepts such as reversing the inequality sign when dividing by a negative number. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation