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.
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