Question

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, x, and grayscale copies, y, that julissa can print to meet her goal?
a. ( 2x + 4y leq 25 )
( x + y geq 8 )
b. ( 4x + 2y geq 25 )
( x + y geq 8 )
c. ( 4x + 2y leq 25 )
( x + y leq 8 )
d. ( 4x + 2y leq 25 )
( x + y geq 8 )

Explanation:

Step1: Analyze time constraint

Let \( x \) be color copies, \( y \) be grayscale. Time for color: \( 4x \), grayscale: \( 2y \). Total time \( \leq 25 \), so \( 4x + 2y \leq 25 \).

Step2: Analyze copy number constraint

Total copies \( x + y \geq 8 \) (needs no fewer than 8).

Step3: Match with options

Check options: Only option B has \( 4x + 2y \geq 25 \)? Wait, no—wait, time is \( \leq 25 \), so \( 4x + 2y \leq 25 \)? Wait, no, recheck: "within 25 minutes" means total time \( \leq 25 \), so \( 4x + 2y \leq 25 \). And "no fewer than 8" means \( x + y \geq 8 \). So option B: \( 4x + 2y \geq 25 \)? No, wait, maybe I misread. Wait, option B: \( 4x + 2y \geq 25 \)? No, let's re-express:

Wait, time for color: 4 mins per, grayscale: 2 mins per. So total time: \( 4x + 2y \leq 25 \) (since within 25 mins). Copies: \( x + y \geq 8 \) (no fewer than 8). So looking at options:

Option B: \( 4x + 2y \geq 25 \)? No, that's time at least 25, which is wrong. Wait, option C: \( 4x + 2y \leq 25 \) and \( x + y \leq 8 \)? No, copies need to be at least 8. Wait, option D: \( 4x + 2y \leq 25 \) and \( x + y \geq 8 \)? Wait, no, option D: \( 4x + 2y \leq 25 \), \( x + y \geq 8 \)? Wait, the options:

A: \( 2x + 4y \leq 25 \), \( x + y \geq 8 \) (wrong time per copy: color is 4, grayscale 2, so A has color as 2, grayscale as 4—wrong).

B: \( 4x + 2y \geq 25 \), \( x + y \geq 8 \) (time at least 25—wrong, should be at most 25).

C: \( 4x + 2y \leq 25 \), \( x + y \leq 8 \) (copies at most 8—wrong, needs at least 8).

D: \( 4x + 2y \leq 25 \), \( x + y \geq 8 \) (time at most 25, copies at least 8—correct). Wait, but earlier I thought B, but no—wait, recheck the options:

Wait, the user's options:

A: \( 2x + 4y \leq 25 \), \( x + y \geq 8 \)

B: \( 4x + 2y \geq 25 \), \( x + y \geq 8 \)

C: \( 4x + 2y \leq 25 \), \( x + y \leq 8 \)

D: \( 4x + 2y \leq 25 \), \( x + y \geq 8 \)

Ah! I see, I misread option D. So time: \( 4x + 2y \leq 25 \) (correct, within 25), copies: \( x + y \geq 8 \) (correct, no fewer than 8). So option D? Wait, no, wait the original options:

Wait, the problem says "no fewer than 8" means \( x + y \geq 8 \), and "within 25 minutes" means \( 4x + 2y \leq 25 \). So option D: \( 4x + 2y \leq 25 \), \( x + y \geq 8 \). Yes, that's correct. Wait, but earlier I thought B, but no—let's re-express:

Color copies: \( x \), time per: 4 → \( 4x \)

Grayscale: \( y \), time per: 2 → \( 2y \)

Total time: \( 4x + 2y \leq 25 \) (within 25 mins)

Total copies: \( x + y \geq 8 \) (no fewer than 8)

So option D: \( 4x + 2y \leq 25 \), \( x + y \geq 8 \). Yes, that's correct. Wait, but earlier I messed up the options. So the correct system is \( 4x + 2y \leq 25 \) and \( x + y \geq 8 \), which is option D? Wait, no, let's check the options again:

Option D: \( 4x + 2y \leq 25 \), \( x + y \geq 8 \) → yes.

Wait, but initially I thought B, but no—probably a misread. So the correct answer is D? Wait, no, wait the options:

Wait, the user's options:

A. \( 2x + 4y \leq 25 \), \( x + y \geq 8 \)

B. \( 4x + 2y \geq 25 \), \( x + y \geq 8 \)

C. \( 4x + 2y \leq 25 \), \( x + y \leq 8 \)

D. \( 4x + 2y \leq 25 \), \( x + y \geq 8 \)

Yes, so D is correct? Wait, no—wait, time for color is 4, grayscale 2, so \( 4x + 2y \), not \( 2x + 4y \). So A has wrong time coefficients. B has time \( \geq 25 \), wrong. C has copies \( \leq 8 \), wrong. D has time \( \leq 25 \) and copies \( \geq 8 \), correct.

Answer:

D. \( 4x + 2y \leq 25 \), \( x + y \geq 8 \)