Answer :
Certainly! Let's break down the steps to solve the problem step-by-step:
### Given Information:
- Lia must work at least 5 hours per week in the restaurant ([tex]$r \geq 5$[/tex]).
- She can work a maximum of 15 hours per week overall ([tex]$r + y \leq 15$[/tex]).
- She wants to earn at least [tex]$120 per week ($[/tex]8r + [tex]$12y \geq 120$[/tex]).
- Hourly wage at the restaurant is [tex]$8 per hour. - Hourly wage for yard work is $[/tex]12 per hour.
We'll answer each part of the question:
### 1. Graph the Inequalities:
To graph the inequalities:
- [tex]$r \geq 5$[/tex]:
Draw a vertical line at [tex]$r = 5$[/tex] and shade to the right (including the line).
- [tex]$r + y \leq 15$[/tex]:
Draw a line passing through points [tex]$(15, 0)$[/tex] and [tex]$(0, 15)$[/tex]. Shade below this line (including the line).
- [tex]$8r + 12y \geq 120$[/tex]:
Rearrange to [tex]$y \geq \frac{120 - 8r}{12} \Rightarrow y \geq 10 - \frac{2r}{3}$[/tex].
Draw this line by finding intercept points, let’s find the intercepts:
- Set [tex]$r = 0 \Rightarrow y = 10$[/tex]
- Set [tex]$y = 0 \Rightarrow $[/tex] [tex]$r = 15$[/tex]
Plot these points [tex]$(0, 10)$[/tex] and [tex]$(15, 0)$[/tex], and shade above this line (including the line).
### 2. The Maximum Number of Hours Lia Can Work at the Restaurant:
To find the maximum number of hours Lia can work at the restaurant ([tex]$r$[/tex]) while still meeting her earnings goal of at least [tex]$120$[/tex], we need to analyze the constraints:
- [tex]$r \geq 5$[/tex] means she must work at least 5 hours at the restaurant.
- [tex]$r + y \leq 15$[/tex] means she wants to work no more than 15 hours in total.
- [tex]$8r + 12y \geq 120$[/tex] represents her earnings goal constraint.
We need to find the feasible values of [tex]$r$[/tex] and [tex]$y$[/tex] that satisfy all three conditions.
Evaluating these constraints, we can isolate the condition involving earnings:
[tex]\[ 8r + 12y \geq 120 \][/tex]
First, express [tex]$y$[/tex] in terms of [tex]$r$[/tex] from [tex]$r + y \leq 15$[/tex]:
[tex]\[ y \leq 15 - r \][/tex]
Substituting [tex]$y = 15 - r$[/tex] into the earnings constraint:
[tex]\[ 8r + 12(15 - r) \geq 120 \][/tex]
[tex]\[ 8r + 180 - 12r \geq 120 \][/tex]
[tex]\[ -4r + 180 \geq 120 \][/tex]
[tex]\[ -4r \geq -60 \][/tex]
[tex]\[ r \leq 15 \][/tex]
Since [tex]$r$[/tex] must also be at least 5:
[tex]\[ 5 \leq r \leq 15 \][/tex]
Thus, the maximum number of hours Lia can work at the restaurant while still achieving her earnings goal is 15 hours.
### 3. The Maximum Amount of Money Lia Can Earn in 1 Week:
Given that Lia can work a maximum of 15 hours at the restaurant (from part 2), we will find her total earnings:
[tex]\[ y = 15 - 15 = 0 \][/tex]
Lia would work all 15 hours at the restaurant and none doing yard work:
Total earnings:
[tex]\[ 8r + 12y = 8(15) + 12(0) = 120 \][/tex]
Thus, the maximum amount of money Lia can earn in one week is [tex]$120. This aligns well with the constraints and ensures that she meets her earnings goal of at least $[/tex]120.
### Given Information:
- Lia must work at least 5 hours per week in the restaurant ([tex]$r \geq 5$[/tex]).
- She can work a maximum of 15 hours per week overall ([tex]$r + y \leq 15$[/tex]).
- She wants to earn at least [tex]$120 per week ($[/tex]8r + [tex]$12y \geq 120$[/tex]).
- Hourly wage at the restaurant is [tex]$8 per hour. - Hourly wage for yard work is $[/tex]12 per hour.
We'll answer each part of the question:
### 1. Graph the Inequalities:
To graph the inequalities:
- [tex]$r \geq 5$[/tex]:
Draw a vertical line at [tex]$r = 5$[/tex] and shade to the right (including the line).
- [tex]$r + y \leq 15$[/tex]:
Draw a line passing through points [tex]$(15, 0)$[/tex] and [tex]$(0, 15)$[/tex]. Shade below this line (including the line).
- [tex]$8r + 12y \geq 120$[/tex]:
Rearrange to [tex]$y \geq \frac{120 - 8r}{12} \Rightarrow y \geq 10 - \frac{2r}{3}$[/tex].
Draw this line by finding intercept points, let’s find the intercepts:
- Set [tex]$r = 0 \Rightarrow y = 10$[/tex]
- Set [tex]$y = 0 \Rightarrow $[/tex] [tex]$r = 15$[/tex]
Plot these points [tex]$(0, 10)$[/tex] and [tex]$(15, 0)$[/tex], and shade above this line (including the line).
### 2. The Maximum Number of Hours Lia Can Work at the Restaurant:
To find the maximum number of hours Lia can work at the restaurant ([tex]$r$[/tex]) while still meeting her earnings goal of at least [tex]$120$[/tex], we need to analyze the constraints:
- [tex]$r \geq 5$[/tex] means she must work at least 5 hours at the restaurant.
- [tex]$r + y \leq 15$[/tex] means she wants to work no more than 15 hours in total.
- [tex]$8r + 12y \geq 120$[/tex] represents her earnings goal constraint.
We need to find the feasible values of [tex]$r$[/tex] and [tex]$y$[/tex] that satisfy all three conditions.
Evaluating these constraints, we can isolate the condition involving earnings:
[tex]\[ 8r + 12y \geq 120 \][/tex]
First, express [tex]$y$[/tex] in terms of [tex]$r$[/tex] from [tex]$r + y \leq 15$[/tex]:
[tex]\[ y \leq 15 - r \][/tex]
Substituting [tex]$y = 15 - r$[/tex] into the earnings constraint:
[tex]\[ 8r + 12(15 - r) \geq 120 \][/tex]
[tex]\[ 8r + 180 - 12r \geq 120 \][/tex]
[tex]\[ -4r + 180 \geq 120 \][/tex]
[tex]\[ -4r \geq -60 \][/tex]
[tex]\[ r \leq 15 \][/tex]
Since [tex]$r$[/tex] must also be at least 5:
[tex]\[ 5 \leq r \leq 15 \][/tex]
Thus, the maximum number of hours Lia can work at the restaurant while still achieving her earnings goal is 15 hours.
### 3. The Maximum Amount of Money Lia Can Earn in 1 Week:
Given that Lia can work a maximum of 15 hours at the restaurant (from part 2), we will find her total earnings:
[tex]\[ y = 15 - 15 = 0 \][/tex]
Lia would work all 15 hours at the restaurant and none doing yard work:
Total earnings:
[tex]\[ 8r + 12y = 8(15) + 12(0) = 120 \][/tex]
Thus, the maximum amount of money Lia can earn in one week is [tex]$120. This aligns well with the constraints and ensures that she meets her earnings goal of at least $[/tex]120.
