Consider the following inequality:

−7≤−34z+5≤−4
Solve the linear inequality for the given variable. Simplify and express your answer in algebraic notation.

To solve the inequality \(-7 \leq -34z + 5 \leq -4\), we'll break it down into two separate inequalities and solve them step by step.

### Step 1: Break the compound inequality into two parts

1. \(-7 \leq -34z + 5\)
2. \(-34z + 5 \leq -4\)

### Step 2: Solve the first inequality \(-7 \leq -34z + 5\)

1. Subtract 5 from both sides:
   \[
   -7 - 5 \leq -34z
   \]
   \[
   -12 \leq -34z
   \]

2. Divide both sides by \(-34\). Since we are dividing by a negative number, the inequality sign will flip:
   \[
   \frac{-12}{-34} \geq z
   \]
   Simplify the fraction:
   \[
   \frac{6}{17} \geq z
   \]
   This can be rewritten as:
   \[
   z \leq \frac{6}{17}
   \]

### Step 3: Solve the second inequality \(-34z + 5 \leq -4\)

1. Subtract 5 from both sides:
   \[
   -34z \leq -4 - 5
   \]
   \[
   -34z \leq -9
   \]

2. Divide both sides by \(-34\). Again, since we are dividing by a negative number, the inequality sign will flip:
   \[
   z \geq \frac{-9}{-34}
   \]
   Simplify the fraction:
   \[
   z \geq \frac{9}{34}
   \]

### Step 4: Combine the results

The solution to the inequality is:
\[
\frac{9}{34} \leq z \leq \frac{6}{17}
\]

This is the solution in algebraic notation.

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the solution change if the inequality was \( -7 \leq -34z + 5 \leq 4 \) instead?
2. Can you solve the inequality \( -2 \leq 5z - 1 \leq 10 \) and express the solution in algebraic notation?
3. What happens to the solution if both sides of the inequality \( z \leq \frac{6}{17} \) are multiplied by 17?
4. How can you check if a specific value of \( z \) satisfies the inequality \( \frac{9}{34} \leq z \leq \frac{6}{17} \)?
5. Can you graph the solution set for \( \frac{9}{34} \leq z \leq \frac{6}{17} \) on a number line?

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

Consider the following inequality:

−7 ≤ −34z + 5 ≤ −4
Solve the linear inequality for the given variable. Simplify and express your answer in algebraic notation.

Learn how to solve the compound linear inequality −7 ≤ −34z + 5 ≤ −4 step-by-step. This problem covers key algebraic concepts, including properties of inequalities and flipping the inequality sign when dividing by a negative number. Suitable for students in Grades 8-10.

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