Question

Jeanne babysits for $6 per hour. She also works as a reading tutor for $10 per hour. She is only allowed to work 20 hours per week. This week, her goal is to make at least $75.
A. Use a system of inequalities to model the scenario above. Let x represent babysitting hours and y represent tutoring hours.
B. Use the model created in part A to create a graph representing Jeanne’s probable income earned and possible number of hours worked this week.
C. Analyze the set of coordinate values that represent solutions for the model created in part A. Choose one of the coordinates within the solution and algebraically prove that the coordinate represents a true solution for the model.

Answer 1

Part a) Use a system of inequalities to model the scenario

Let

x-------> represent babysitting hours

y-------> represent tutoring hours  

we know that

$x+y \leq20$ --------> inequality $1$  

$6x+10y \geq 75$ --------> inequality $2$

Part b) Use the model created in part A to create a graph representing Jeanne’s probable income earned and possible number of hours worked this week

we know that

$x+y \leq20$ --------> inequality $1$

This inequality represent Jeanne’s possible number of hours worked this week

Using a graph tool

the solution of this inequality is the shaded area

see the attached figure N $1$

$6x+10y \geq 75$ --------> inequality $2$  

This inequality represent Jeanne’s probable income earned this week

Using a graph tool

the solution of this inequality is the shaded area

see the attached figure N $2$

Part c) Analyze the set of coordinate values that represent solutions for the model created in part A. Choose one of the coordinates within the solution and algebraically prove that the coordinate represents a true solution for the model

we have the system of inequalities

$x+y \leq20$ --------> inequality $1$

$6x+10y \geq 75$ --------> inequality $2$

the solution of the system of inequalities is the shaded area

using a graph tool

see the attached figure N  $3$

Let

$A(10,10)$ ------> see the attached figure N  $3$

The point belongs to the shaded area, so it is a system solution.

If the point is a solution, it must satisfy both inequalities.

Prove algebraically

$A(10,10)$

substitute the value of x and y in the inequalities

inequality $1$

$10+10 \leq20$

$20 \leq20$ ------> is ok

inequality $2$

$6*10+10*10 \geq 75$

$160 \geq 75$ -------> is ok

the point A satisfy both inequalities

therefore

the point A is a solution of the system