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Difficulty: EasySystems of Linear Inequalities in Two Variables

A graphic designer allocates their weekly work time between designing layouts, which takes dd hours, and client consultations, which takes cc hours. The designer works a maximum of 35 hours per week. Additionally, the designer spends at least twice as much time designing layouts as they do in client consultations. Which of the following systems of inequalities represents this situation?

  1. d+c≤35d≥2c\begin{aligned} d + c &\leq 35 \\ d &\geq 2c \end{aligned}Answer
  2. B
    d+c≥35d≤2c\begin{aligned} d + c &\geq 35 \\ d &\leq 2c \end{aligned}
  3. C
    d+c≤35c≥2d\begin{aligned} d + c &\leq 35 \\ c &\geq 2d \end{aligned}
  4. D
    d+c≥35c≤2d\begin{aligned} d + c &\geq 35 \\ c &\leq 2d \end{aligned}

Answer

The system with the inequalities d+c≤35d + c \leq 35 and d≥2cd \geq 2c
The system containing the inequalities d+c≤35d + c \leq 35 and d≥2cd \geq 2c is correct. The constraint 'works a maximum of 35 hours per week' means the sum of the hours, d+cd + c, must be less than or equal to 35 (d+c≤35d + c \leq 35). The constraint 'spends at least twice as much time designing layouts as they do in client consultations' means that design hours, dd, must be greater than or equal to twice the consultation hours, cc (d≥2cd \geq 2c).

Step-by-Step Solution

1
Identify the inequality representing the limit on total weekly hours.
d+c≤35d + c \leq 35
The phrase 'works a maximum of 35 hours per week' means the total hours spent on designing (dd) and consulting (cc) cannot exceed 35.
2
Identify the inequality representing the relative relationship between designing and consulting hours.
d≥2cd \geq 2c
The phrase 'at least twice as much time designing layouts as they do in client consultations' means the design hours (dd) must be greater than or equal to two times the consultation hours (2c2c).
3
Combine the individual inequalities into a system.
The system consists of d+c≤35d + c \leq 35 and d≥2cd \geq 2c.
Both conditions must be satisfied simultaneously to represent the designer's weekly schedule constraint.

Key Concept

Systems of Linear Inequalities in Two Variables
Estimated Time:1m 0s
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Systems of Linear Inequalities in Two Variables

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