Question

Using the properties of inequality, justify both steps in solving the inequality 25m−7≤17
.(4 points)

Answer 1

To solve the inequality \( 25m - 7 \leq 17 \), we will justify each step of the solution using properties of inequalities.

### Step 1: Add 7 to both sides
We start with the inequality:
\[ 25m - 7 \leq 17 \]

To eliminate the constant term (-7) on the left side, we add 7 to both sides of the inequality:
\[ 25m - 7 + 7 \leq 17 + 7 \]
\[ 25m \leq 24 \]

**Justification: Property of Addition**
The property states that if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \). Here, \( c \) is 7.

### Step 2: Divide both sides by 25
Now we have:
\[ 25m \leq 24 \]

To isolate \( m \), we divide both sides by 25:
\[ \frac{25m}{25} \leq \frac{24}{25} \]
\[ m \leq \frac{24}{25} \]

**Justification: Property of Division**
The property states that if \( a \leq b \), and \( c > 0 \), then \( \frac{a}{c} \leq \frac{b}{c} \). Here, \( c \) is 25, which is a positive number.

### Conclusion
Thus, the final solution to the inequality is:
\[ m \leq \frac{24}{25} \]

We justified both steps using the properties of addition and division to maintain the integrity of the inequality.