Just answer the last one. Please 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, €, she can use the boat: Emilia’s Model X ≤ 6 10 12 14 Are Emilia’s inequality and number line correct? Explain your reasoning by completing the statements. Click the arrows to choose an answer from each menu: Emilia’s inequality, € 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_ should be shaded to the left is correctly shaded to the right of the endpoint:
The Correct Answer and Explanation is:
Corrected Answer:
Emilia’s inequality, x ≤ 6, is correct because the number of hours is at most 6.
Emilia’s number line is incorrect.
The endpoint should be filled in and the number line should be shaded to the left.
Explanation
Emilia is renting a boat for up to 6 hours for $24. The phrase “up to 6 hours” means she can use the boat for any time from 0 hours up to and including 6 hours, but not more than 6 hours.
The inequality x ≤ 6 accurately represents this situation. In mathematics, “≤” means “less than or equal to.” Here, x stands for the number of hours Emilia uses the boat. So x ≤ 6 means she can use the boat for 6 hours or any amount of time less than that—such as 1, 3.5, or 5.9 hours. Therefore, Emilia’s inequality is correct because it reflects the idea of a maximum allowed time.
However, her number line representation has an issue. For an inequality like x ≤ 6, the number line should have:
- A filled-in (solid) circle at 6, because 6 is included in the allowed values.
- A shaded region to the left of 6, representing all values less than or equal to 6.
If Emilia’s number line instead had an open circle at 6 or shaded to the right, it would incorrectly suggest that values greater than 6 are allowed or that 6 is not included, both of which would be wrong.
In conclusion, while Emilia correctly wrote the inequality, her number line must be corrected to reflect the meaning of “x ≤ 6”: a solid point at 6 and shading to the left, capturing all values from 0 up to and including 6.
