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1.4<0.6–0.8(6z–2)≤3.4
Write your answer as a compound inequality with integers, decimals, or proper or improper fractions in simplest form.



We are given the inequality:

\[
1.4 < 0.6 - 0.8(6z - 2) \leq 3.4
\]

### Step 1: Expand the expression inside the inequality
First, distribute the \(-0.8\) across the terms inside the parentheses:

\[
-0.8(6z - 2) = -0.8 \cdot 6z + (-0.8 \cdot -2) = -4.8z + 1.6
\]

Now the inequality becomes:

\[
1.4 < 0.6 - 4.8z + 1.6 \leq 3.4
\]

### Step 2: Simplify the constants
Combine the constants on the right side of the inequality:

\[
0.6 + 1.6 = 2.2
\]

Thus, the inequality becomes:

\[
1.4 < 2.2 - 4.8z \leq 3.4
\]

### Step 3: Solve the inequality
We'll now break this into two inequalities:

1. \( 1.4 < 2.2 - 4.8z \)
2. \( 2.2 - 4.8z \leq 3.4 \)

#### Inequality 1: \( 1.4 < 2.2 - 4.8z \)

Subtract \(2.2\) from both sides:

\[
1.4 - 2.2 < -4.8z
\]
\[
-0.8 < -4.8z
\]

Now, divide both sides by \(-4.8\). When dividing by a negative number, reverse the inequality sign:

\[
\frac{-0.8}{-4.8} > z
\]
\[
\frac{0.8}{4.8} > z \quad \Rightarrow \quad \frac{1}{6} > z
\]

So, \( z < \frac{1}{6} \).

#### Inequality 2: \( 2.2 - 4.8z \leq 3.4 \)

Subtract \(2.2\) from both sides:

\[
-4.8z \leq 3.4 - 2.2
\]
\[
-4.8z \leq 1.2
\]

Now, divide both sides by \(-4.8\) and reverse the inequality:

\[
z \geq \frac{1.2}{-4.8}
\]
\[
z \geq -\frac{1}{4}
\]

### Step 4: Combine the results

From the two inequalities, we have:

\[
-\frac{1}{4} \leq z < \frac{1}{6}
\]

This is the solution to the inequality in compound form.

Would you like further details on any of the steps?

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**Related Questions:**
1. How do you solve compound inequalities step by step?
2. What happens when you divide or multiply by a negative in inequalities?
3. Can compound inequalities have more than one solution?
4. How do you graph compound inequalities on a number line?
5. What are the differences between "and" and "or" compound inequalities?

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

1.4 < 0.6 - 0.8(6z - 2) ≤ 3.4. Write your answer as a compound inequality with integers, decimals, or proper or improper fractions in simplest form.

Learn how to solve the compound inequality 1.4 < 0.6 - 0.8(6z - 2) ≤ 3.4 step by step. This problem involves key algebraic concepts such as the distributive property and inequality rules. Suitable for students in Grades 8-10.

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