- Solve the inequality \(3c \geq 12\) by dividing both sides by 3.
- Obtain \(c \geq 4\) as the correct result.
- Compare with Devin's result (\(c \geq 9\)) and identify that Devin incorrectly used subtraction instead of division.
- Conclude that option (a) is the correct choice because the number 9 should be 4.
- The final answer is \(\boxed{a}\).
Explanation
- Solving the Inequality and Analyzing Devin's Work
To evaluate whether Devin's work is correct, we analyze the provided inequality:
\[3c \geq 12\]
The goal is to solve for \(c\). To isolate \(c\), we need to apply the division property of inequalities. This property states that if we divide both sides of an inequality by the same positive constant, the direction of the inequality remains unchanged. Dividing both sides by 3 gives:
\[\frac{3c}{3} \geq \frac{12}{3}\]
Simplifying the fractions, we get:
\[c \geq 4\]
Comparing this result to Devin's stated answer of \(c \geq 9\), it is clear that Devin made an error. Devin likely subtracted 3 from 12 instead of dividing 12 by 3 (i.e., \(12 - 3 = 9\)), which is an incorrect operation for solving this type of linear inequality. Therefore, the correct result is \(c \geq 4\).
- Comparing Results and Selecting the Correct Option
Based on our analysis, the correct solution to the inequality \(3c \geq 12\) is \(c \geq 4\). Comparing this to the given options:
ANo, Devin made a mistake. The \(9\) should be \(4\).
BNo, Devin made a mistake. The \(9\) should be \(15\).
CNo, Devin made a mistake. The \(9\) should be \(36\).
DYes, Devin's work is correct.
The correct conclusion is that Devin made a mistake, and the value should be \(4\). Thus, option (a) is the correct choice.
Examples
In everyday life, we often use inequalities to manage budgets or resources. For instance, if you have a budget of \$12 to buy notebooks that cost \$3 each, the inequality \(3c \leq 12\) (where \(c\) is the number of notebooks) helps you determine how many you can afford. Similarly, understanding the correct algebraic steps to solve \(3c \geq 12\) ensures that when you are calculating thresholds for resource allocation, you arrive at the right conclusion, helping you make informed decisions about your purchases or project requirements.