Using the properties of inequality, justify both steps in solving the inequality 2/5 m −7≤17.(4 points)

To solve the inequality \( \frac{2}{5} m - 7 \leq 17 \), we will break down the steps and justify them using properties of inequalities.

### Step 1: Isolate the term with the variable
We start with the inequality:
\[
\frac{2}{5} m - 7 \leq 17
\]
To isolate the term with \( m \), we need to eliminate the constant term \(-7\) on the left side. We do this by adding \( 7 \) to both sides of the inequality.

**Justification:**
Adding the same number to both sides of an inequality maintains the inequality. Therefore:
\[
\frac{2}{5} m - 7 + 7 \leq 17 + 7
\]
This simplifies to:
\[
\frac{2}{5} m \leq 24
\]

### Step 2: Eliminate the coefficient of \( m \)
Next, we want to solve for \( m \) by eliminating the coefficient \(\frac{2}{5}\). We achieve this by multiplying both sides of the inequality by the reciprocal of \(\frac{2}{5}\), which is \(\frac{5}{2}\).

**Justification:**
Multiplying both sides of an inequality by a positive number does not change the direction of the inequality. Thus:
\[
m \leq 24 \cdot \frac{5}{2}
\]
Calculating the right side:
\[
m \leq 12 \cdot 5
\]
\[
m \leq 60
\]

### Final Result
The solution to the inequality \( \frac{2}{5} m - 7 \leq 17 \) is:
\[
m \leq 60
\]

In summary, both steps in solving the inequality were justified by the properties of inequalities: adding a constant does not change the direction of the inequality, and multiplying by a positive number also maintains the inequality.