using the properties of inequality, justify both steps in solving the inequality 2/5m−7≤17.
Using the Properties of Inequality to Solve \frac{2}{5}m - 7 \leq 17
Step 1: Isolating the Term with the Variable
To solve the inequality \frac{2}{5}m - 7 \leq 17, we need to isolate the term involving the variable m. The inequality currently has the term -7 on the left-hand side. Our goal is to remove this term to get one step closer to isolating m. We do this by adding 7 to both sides of the inequality.
- Properties Used: This step utilizes the Addition Property of Inequality, which states that adding the same number to both sides of an inequality does not change the inequality’s direction.
\frac{2}{5}m - 7 + 7 \leq 17 + 7
Simplifying both sides, we have:
\frac{2}{5}m \leq 24
After this step, \frac{2}{5}m is isolated on the left side, which simplifies the inequality for the next step.
Step 2: Solving for the Variable
Now, our task is to solve for m. The term \frac{2}{5}m indicates that m is being multiplied by \frac{2}{5}. To isolate m, we need to perform the inverse operation of multiplication by \frac{2}{5}, which is multiplication by its reciprocal, \frac{5}{2}.
- Properties Used: This step uses the Multiplication Property of Inequality, which states that multiplying both sides of an inequality by a positive number does not change the direction of the inequality. It’s important to note that this property holds true only for positive numbers, and if we were multiplying by a negative number, the inequality would reverse direction.
\frac{5}{2} \times \frac{2}{5}m \leq 24 \times \frac{5}{2}
Simplifying the left-hand side:
m \leq 24 \times \frac{5}{2}
Calculating the right-hand side:
m \leq \frac{120}{2}
m \leq 60
Thus, the solution to the inequality is m \leq 60.
Summary
By using the properties of inequality, we have justified each step in solving \frac{2}{5}m - 7 \leq 17. First, by applying the Addition Property of Inequality, we moved the constant term across the inequality. Second, through the Multiplication Property of Inequality, we isolated the variable m to determine its possible values. Therefore, the solution to the given inequality is m \leq 60, meaning m can be any number up to and including 60.
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