–
4≤2–
p
3
<
2
5
Write your answer as a compound inequality with integers, decimals, or proper or improper fractions in simplest form.

We are given the inequality:

\[
4 \leq \frac{2 - p}{3} < \frac{2}{5}
\]

We need to solve this compound inequality for \( p \).

### Step 1: Break it into two inequalities.
We split the compound inequality into two parts:
1. \( 4 \leq \frac{2 - p}{3} \)
2. \( \frac{2 - p}{3} < \frac{2}{5} \)

### Step 2: Solve the first inequality.
#### Inequality 1:
\[
4 \leq \frac{2 - p}{3}
\]

Multiply both sides by 3 to eliminate the denominator:
\[
4 \times 3 \leq 2 - p
\]
\[
12 \leq 2 - p
\]

Subtract 2 from both sides:
\[
12 - 2 \leq -p
\]
\[
10 \leq -p
\]

Multiply both sides by \(-1\) and reverse the inequality sign:
\[
-10 \geq p
\]
\[
p \leq -10
\]

### Step 3: Solve the second inequality.
#### Inequality 2:
\[
\frac{2 - p}{3} < \frac{2}{5}
\]

Multiply both sides by 3 to eliminate the denominator:
\[
2 - p < \frac{6}{5}
\]

Subtract 2 from both sides:
\[
2 - p - 2 < \frac{6}{5} - 2
\]
\[
-p < \frac{6}{5} - \frac{10}{5}
\]
\[
-p < -\frac{4}{5}
\]

Multiply both sides by \(-1\) and reverse the inequality sign:
\[
p > \frac{4}{5}
\]

### Step 4: Combine the two inequalities.
We now have two results:
\[
p \leq -10 \quad \text{and} \quad p > \frac{4}{5}
\]

However, these two inequalities contradict each other because \( p \) cannot be both less than or equal to \(-10\) and greater than \(\frac{4}{5}\) simultaneously. Therefore, **there is no solution** to this compound inequality.

Would you like further details on this, or do you have any questions?

### Related Questions:
1. How do you solve compound inequalities with different directions?
2. What happens when there's no overlap between two inequality solutions?
3. How do you handle inequalities involving fractions?
4. Why do inequality signs reverse when multiplying by negative numbers?
5. What are common methods for solving inequalities with absolute values?

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

4≤(2-p)/3<2/5. 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 4≤(2-p)/3<2/5. This step-by-step solution walks through splitting the inequality, eliminating fractions, and solving for p. Ideal for students in Grades 8-10 covering topics like inequalities and fraction manipulation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation