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Emilia pays $24 to rent a boat at the lake for up to 6 hours. She constructs the inequality and number line below to represent the number of hours, $x$, she can use the boat.



Emilia's Model



Are Emilia's inequality and number line correct? Explain your reasoning by completing the statements.



Emilia's inequality, $x \leq 6$, is [Choose...] because the number of hours is [Choose...].



Emilia's number line is [Choose...]. The endpoint [Choose...] filled in and the number line [Choose...] of the endpoint.

Verified Answer :

  • The constraint for the number of hours \(x\) is \(0 \leq x \leq 6\) because time must be non-negative and the limit is 6 hours.
  • Emilia's inequality \(x \leq 6\) is incomplete because it ignores that time is non-negative (\(x \geq 0\)).
  • The number line is incomplete; it should show a closed interval from 0 to 6.
  • The endpoints at 0 and 6 should be filled in and the line should be shaded between the two points.

Explanation

  1. Analyzing the Problem Constraints

The problem asks us to evaluate whether Emilia's mathematical model accurately represents the constraint that she can rent a boat for up to 6 hours. The variable \(x\) represents the number of hours she uses the boat.

In a real-world context, time cannot be negative. Therefore, the number of hours \(x\) must be at least 0. This gives us the constraint \(x \geq 0\). Additionally, the problem specifies that she can rent the boat for "up to 6 hours," meaning the maximum amount of time is 6 hours, or \(x \leq 6\). Combining these two requirements, the complete range of valid hours is \(0 \leq x \leq 6\).

  1. Evaluating the Inequality

Emilia's proposed inequality is \(x \leq 6\). While this correctly identifies the upper bound of the rental time, it is incomplete because it does not account for the physical reality that time cannot be negative. Since the number of hours must be at least 0, the complete inequality is \(0 \leq x \leq 6\). Therefore, Emilia's inequality is incomplete because the number of hours must be non-negative.

  1. Evaluating the Number Line Representation

Regarding the number line, the representation of \(x \leq 6\) typically involves an endpoint at 6. Since the condition is "less than or equal to" (\(\leq\)), the endpoint at 6 should be a closed (filled) circle. The line should be shaded to the left of the endpoint to represent all values less than 6. However, because the number of hours must also be greater than or equal to 0, the correct representation should be a closed interval starting at 0 and ending at 6, with a filled circle at both 0 and 6, and the line shaded between them. Consequently, Emilia's number line is incomplete. It should show the interval [0, 6], with both endpoints at 0 and 6 filled in, and the line shaded between the two points.

Examples

Imagine you are setting a timer for an oven. If you say you want the oven on for "no more than 60 minutes," you implicitly understand that the timer cannot start at a negative number of minutes. Similarly, when tracking resources or time limits, identifying the realistic bounds (like zero for time or weight) is just as important as identifying the maximum capacity. This ensures that models accurately reflect real-life possibilities rather than just abstract mathematical ranges.

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