Q32. - Algebra 2

Question

Bryan Clark makes \(10 an hour cutting grass and \)12 an hour for raking leaves. He cannot work more than 15 hours per week. Graph two inequalities that Bryan can use to determine how many hours he needs to work at each job if he wants to earn at least $120 per week.

Step-by-Step Solution

Answer

The shaded region represents numbers of hours he needs to work at each job. Where x-axis represents number of hours spent in cutting grass and y-axis represents number of hours spent in raking leaves.

Step-1 – Define the variable

Bryan Clark makes $10 an hour cutting grass and $12 an hour for raking leaves. 

He cannot work more than 15 hours per week. 

He wants to earn at least $120 per week.

Let the x represents number of hours spent in cutting grass and y represents number of hours spent in raking leaves.

Step-2 – Construct system of inequalities

The system of inequalities that represent the data are provided below.

He cannot work more than 15 hours per week. 

$x+y<15$

Bryan Clark makes $10 an hour cutting grass and $12 an hour for raking leaves. 

He wants to earn at least $120 per week.

$10x+12y\ge 120$

 

Therefore, the system of inequalities is provided below.

 $\begin{array}{c}10x+12y\ge 120\\ x+y<15\end{array}$

Step-3 &ndash; Apply the concept of graphing the inequality

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid. 

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality. 

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step-4 &ndash; Interpret the sign of the inequality

Consider the inequality provided below.

$10x+12y\ge 120$

The inequality contains the sign of greater than or equal to.

Therefore, the boundary line will be solid.

Next, consider the inequalities.

$x+y<15$

The inequality contains the sign of less than.

Therefore, the boundary line will be dashed.

Step-5 &ndash; Graph the inequalities

Graph the inequalities $10x+12y\ge 120$ and $x+y<15$ on same plane and shade the region.

The corresponding equation is $10x+12y=120$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$. 

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

 $\begin{array}{c}10\left(0\right)+12\left(0\right)\ge 120\\ 0\ge 120\end{array}$

This is false.

Therefore, shade the region not containing the point $\left(0,0\right)$.

Draw the line $x+y=15$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$. 

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

 $\begin{array}{c}0+0<15\\ 0<15\end{array}$

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Thus, the shaded region is provided below.

The region 1 and region 3 corresponds to inequality $x+y<15$.

The region 2 and region 3 corresponds to inequality $10x+12y\ge 120$.

Step-6 &ndash; Shade the common region

The region common to both the inequalities $10x+12y\ge 120$ and $x+y<15$ is Region 3.

Hence, the shaded region represents numbers of hours he needs to work at each job. Where x-axis represents number of hours spent in cutting grass and y-axis represents number of hours spent in raking leaves.