Question

john has to fix at least 12 watches in 40 hours. digital watches (x) take him 2 hours to fix and analog watches (y) take him 6 hours to fix.

complete the inequalities below to represent this situation:

\\x + y \ge ?\\
\\ x + y \le 40\\

Explanation:

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<reasoning>

Define variables and constraints

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &x = \text{number of digital watches}\\ &y = \text{number of analog watches} \end{aligned}$$

\]

Model the watch quantity constraint

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Total watches} \ge 12\\ &x + y \ge 12 \end{aligned}$$

\]

Model the time constraint

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Time for digital watches} = 2x\\ &\text{Time for analog watches} = 6y\\ &\text{Total time} \le 40\\ &2x + 6y \le 40 \end{aligned}$$

\]
</reasoning>

<answer>
John has to fix at least 12 watches in 40 hours. Digital watches (\(x\)) take him 2 hours to fix and analog watches (\(y\)) take him 6 hours to fix.

Complete the inequalities below to represent this situation:

\(x + y \ge\) <blank>12</blank>

<blank>2</blank>\(x +\) <blank>6</blank>\(y \le 40\)
</answer>

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</post_analysis>

Answer:

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<reasoning>

Define variables and constraints

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &x = \text{number of digital watches}\\ &y = \text{number of analog watches} \end{aligned}$$

\]

Model the watch quantity constraint

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Total watches} \ge 12\\ &x + y \ge 12 \end{aligned}$$

\]

Model the time constraint

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Time for digital watches} = 2x\\ &\text{Time for analog watches} = 6y\\ &\text{Total time} \le 40\\ &2x + 6y \le 40 \end{aligned}$$

\]
</reasoning>

<answer>
John has to fix at least 12 watches in 40 hours. Digital watches (\(x\)) take him 2 hours to fix and analog watches (\(y\)) take him 6 hours to fix.

Complete the inequalities below to represent this situation:

\(x + y \ge\) <blank>12</blank>

<blank>2</blank>\(x +\) <blank>6</blank>\(y \le 40\)
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Systems of Linear Inequalities"
]
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