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Eve knits scarves for her online store. This week she has already knitted 9 scarves. There are 2 days left in the work week. She set a goal for herself to knit at least 15 scarves and at the most 19 scarves this week. Eve wants to know how many scarves she should knit each day to meet her goal.



Which inequality represents the given situation?


A$15 \leq 2x - 9 \geq 19$
B$15 \geq 2x + 9 \geq 19$
C$15 \leq 2x + 9 \leq 19$
D$15 \leq 2x - 9 \leq 19$

Verified Answer :

  • Eve starts with 9 scarves and knits \(2x\) more, totaling \(9 + 2x\).
  • Her goal is between 15 and 19 inclusive, which is expressed as \(15 \leq 2x + 9 \leq 19\).
  • Comparing this to the options, the correct choice is: \(\boxed{15 \leq 2x + 9 \leq 19}\).

Explanation

  1. Analyze the problem variables and expression

Eve has already completed 9 scarves and plans to work for 2 more days. If she knits \(x\) scarves per day, the total number of new scarves she will knit in these 2 days is \(2x\). Therefore, her total number of scarves for the week will be the sum of the scarves already completed and the new ones, which is represented by the expression \(9 + 2x\).

  1. Construct the compound inequality

Eve's goal is to knit at least 15 scarves and at most 19 scarves. This means the total number of scarves, \(9 + 2x\), must satisfy two conditions simultaneously: it must be greater than or equal to 15, and less than or equal to 19. This is written as:
\[15 \leq 9 + 2x \leq 19\]
Alternatively, we can express the total as \(2x + 9\) without changing the inequality's meaning:
\[15 \leq 2x + 9 \leq 19\]

  1. Select the correct inequality

Now we compare our derived inequality \(15 \leq 2x + 9 \leq 19\) to the provided options:

  • \(15 \leq 2x - 9 \geq 19\) (Incorrect)
  • \(15 \geq 2x + 9 \geq 19\) (Incorrect)
  • \(15 \leq 2x + 9 \leq 19\) (Matches our result)
  • \(15 \leq 2x - 9 \leq 19\) (Incorrect)

The correct inequality that represents Eve's goal is \(15 \leq 2x + 9 \leq 19\).

Examples

This type of inequality is used in project management and resource planning. For example, if a team has already completed a portion of a project (like writing 9 chapters of a book) and needs to finish within a certain timeframe with a target range (like finishing between 15 and 19 chapters total), they can use linear inequalities to determine how much work needs to be done each day. By solving \(15 \leq 2x + 9 \leq 19\), the team finds that \(6 \leq 2x \leq 10\), which means they must complete between 3 and 5 chapters each day.

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