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

Answer 1

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.