Select the correct answer.
Julissa is printing out copies for a work training. It takes 4 minutes to print a color copy, and it takes 2 minutes to print a grayscale copy. She needs to print no fewer than 8 copies within 25 minutes.
Which system of inequalities represents the number of color copies, [tex]$x$[/tex], and grayscale copies, [tex]$y$[/tex], that Julissa can print to meet her goal?
A. [tex]$2 x+4 y \leq 25$[/tex]
[tex]$x+y \geq 8$[/tex]
B. [tex]$4 x+2 y \geq 25$[/tex]
[tex]$x+y \geq 8$[/tex]
C. [tex]$4 x+2 y \leq 25$[/tex]
[tex]$x+y \leq 8$[/tex]
D. [tex]$4 x+2 y \leq 25$[/tex]
[tex]$x+y \geq 8$[/tex]
Answer (1)
Define variables: x = number of color copies, y = number of grayscale copies.
Write the inequality for the total time: 4 x + 2 y ≤ 25 .
Write the inequality for the total number of copies: x + y ≥ 8 .
Select the correct system of inequalities: 4 x + 2 y ≤ 25 , x + y ≥ 8 .
Explanation
Understanding the Problem Let's break down this word problem to create a system of inequalities that represents Julissa's printing situation. We need to consider the time it takes to print each type of copy and the total number of copies she needs to print.
Defining Variables Let x represent the number of color copies and y represent the number of grayscale copies.
Setting up the Time Inequality Since it takes 4 minutes to print a color copy and 2 minutes to print a grayscale copy, the total time to print x color copies and y grayscale copies is 4 x + 2 y minutes. Julissa needs to print within 25 minutes, so we have the inequality: 4 x + 2 y ≤ 25
Setting up the Copy Number Inequality Julissa needs to print no fewer than 8 copies, meaning the total number of copies, x + y , must be greater than or equal to 8. This gives us the inequality: x + y ≥ 8
The System of Inequalities Combining these two inequalities, we get the system: 4 x + 2 y ≤ 25 x + y ≥ 8
Finding the Correct Answer Now, let's compare our system of inequalities to the answer choices. Option D matches our system: 4 x + 2 y ≤ 25 x + y ≥ 8
Examples
Imagine you're baking cookies for a bake sale. You want to make at least 2 dozen cookies (24 cookies) but only have 3 hours (180 minutes) to bake. If chocolate chip cookies take 5 minutes per dozen to prepare and bake, and oatmeal cookies take 7 minutes per dozen, you can use a system of inequalities to figure out how many dozens of each type of cookie to bake to meet your goals. This helps manage your time and resources effectively, ensuring you have enough cookies for the sale without exceeding your baking time. Let x be the number of chocolate chip cookie dozens and y be the number of oatmeal cookie dozens. Then x + y ≥ 2 and 5 x + 7 y ≤ 180 .
