Question
Using the properties of inequality, justify both steps in solving the inequality 2/5m - 7 less than or equal to 17
Answers
GPT-4o mini
To solve the inequality \( \frac{2}{5}m - 7 \leq 17 \), we can follow these steps while justifying each step using properties of inequalities:
**Step 1: Add 7 to both sides.**
Justification: When you add the same value to both sides of an inequality, the inequality sign remains the same.
Starting with the original inequality:
\[
\frac{2}{5}m - 7 \leq 17
\]
Add 7 to both sides:
\[
\frac{2}{5}m - 7 + 7 \leq 17 + 7
\]
This simplifies to:
\[
\frac{2}{5}m \leq 24
\]
---
**Step 2: Multiply both sides by \( \frac{5}{2} \).**
Justification: When you multiply both sides of an inequality by a positive number, the inequality sign remains the same.
Multiply both sides by \( \frac{5}{2} \):
\[
\frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2}
\]
The left side simplifies as follows:
\[
m \leq \frac{120}{2}
\]
This simplifies to:
\[
m \leq 60
\]
---
**Conclusion**: The solution to the inequality \( \frac{2}{5}m - 7 \leq 17 \) is \( m \leq 60 \). Both steps maintain the inequality correctly, as justified above.
**Step 1: Add 7 to both sides.**
Justification: When you add the same value to both sides of an inequality, the inequality sign remains the same.
Starting with the original inequality:
\[
\frac{2}{5}m - 7 \leq 17
\]
Add 7 to both sides:
\[
\frac{2}{5}m - 7 + 7 \leq 17 + 7
\]
This simplifies to:
\[
\frac{2}{5}m \leq 24
\]
---
**Step 2: Multiply both sides by \( \frac{5}{2} \).**
Justification: When you multiply both sides of an inequality by a positive number, the inequality sign remains the same.
Multiply both sides by \( \frac{5}{2} \):
\[
\frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2}
\]
The left side simplifies as follows:
\[
m \leq \frac{120}{2}
\]
This simplifies to:
\[
m \leq 60
\]
---
**Conclusion**: The solution to the inequality \( \frac{2}{5}m - 7 \leq 17 \) is \( m \leq 60 \). Both steps maintain the inequality correctly, as justified above.