To solve the inequality \( \frac{2}{5}m - 7 < 17 \), we can use the properties of inequality to isolate the variable \( m \) on one side. Here are the steps justifying each action taken:
Step 1: Add 7 to both sides of the inequality.
Justification: We want to eliminate the -7 on the left side to isolate the term with \( m \). According to the Properties of Inequality, we can add or subtract the same number from both sides of the inequality without changing the direction of the inequality. This property is similar to the addition property of equality. Therefore, adding 7 to both sides maintains the inequality.
\( \frac{2}{5}m - 7 + 7 < 17 + 7 \)
Simplified, it becomes:
\( \frac{2}{5}m < 24 \)
Step 2: Multiply both sides of the inequality by the reciprocal of \( \frac{2}{5} \), which is \( \frac{5}{2} \).
Justification: To solve for \( m \), we need to eliminate the coefficient of \( \frac{2}{5} \). Multiplying by the reciprocal is the same as dividing by the original number, which will cancel out the coefficient, leaving \( m \) alone on one side. The Properties of Inequality allow us to multiply or divide both sides of the inequality by a positive number without changing the direction of the inequality. It is important to note that if we were to multiply or divide by a negative number, the inequality sign would need to be reversed. Here we are multiplying by \( \frac{5}{2} \), which is positive, so the sign remains the same.
\( \frac{5}{2} \times \frac{2}{5}m < 24 \times \frac{5}{2} \)
Simplified, it becomes:
\( m < 60 \)
Therefore, by using these two steps and the properties of inequality, we have correctly solved the original inequality and found that \( m \) must be less than 60.
Using the properties of inequality, justify both steps in solving the inequality 2/5 m-7<17
1 answer
1 answer