Question

Kishi makes$\mathit{\backslash (}\mathbf{7}$ an hour working at the grocery store and $\mathit{\backslash )}\mathbf{10}$an hour delivering newspapers. She cannot work more than $\mathbf{20}\mathit{h}\mathit{o}\mathit{u}\mathit{r}\mathit{s}$ per week. Graph two inequalities that Kishi can use to determine how many hours she needs to work at each job if she wants to earn at least$\mathit{\$}\mathbf{90}$per week.

Short answer

The graph of the two inequalities that Kishi can use to determine how many hours she needs to work at each job if she wants to earn at least $\$90$per week is:

Step-by-step solution

Step 1. Determine the two inequalities.

Let Kishi works for $x$ hours at the grocery store and Kishi works for $y$hours

delivering newspapers.

It is given that Kishi cannot work more than $20$ hours per week.

Therefore, it implies that:

role="math" localid="1648200959780" $x+y\le 20...\left(1\right)$

It is given that Kishi makes $\&7$an hour working at the grocery. Therefore, the money earned while working for $x$hours at the grocery store is given by $\$\left(7x\right)$.

It is given that Kishi makes $\&10$an hour in delivering newspapers. Therefore, the money earned while working for $y$ hours in delivering newspapers is given by $\$\left(10y\right)$.

Therefore, the total money earned by Kishi is given $\$(7x+10y)$.

It is given that Kishi wants to earn at least $\&90$per week.

Therefore, it implies that:

$7x+10y\ge 90\text{...}\left(2\right)$

Therefore, the two inequalities are:

role="math" localid="1648200797493" $x+y\le 20\text{...}\left(1\right)$

$7x+10y\ge 90...\left(2\right)$

Step 2. Draw the graph of the inequality x+y≤20.

Convert the inequality $x+y\le 20$ into equality that is convert the inequality into an equation.

Therefore, it is obtained that: $x+y=20$

Therefore, the equation of the boundary is $x+y=20$. The inequality $x+y\le 20$ has equal to sign, therefore the boundary is included in the solution. Therefore, the boundary is denoted by solid lines.

Draw the graph of the boundary line $x+y=20$.

Substitute 0 for $x$ and find the value of $y$.

$\begin{array}{l}x+y=20\\ 0+y=20\\ y=20\end{array}$

Therefore, one of the points is

Substitute 0 for $x$ and find the value of $y$.

$\begin{array}{l}x+y=20\\ x+0=20\\ x=20\end{array}$

Therefore the other point is $(20,0)$

Therefore, draw the graph of the boundary line $x+y=20$ by drawing a line passing through the points $(0,20)$ and $(20,0)$.

Now to draw the graph of the inequality $x+y\le 20$, take any point which is not on the line $x+y=20$ and substitute it in the inequality $x+y\le 20$. If the condition obtained is true, then shade the region towards that point and if the condition obtained is false, then shade the region away from the point.

Let the point be $(0,0)$ and the point $(0,0)$ is not on the line $x+y=20$.

Substitute the point in the inequality $x+y\le 20$.

$\begin{array}{l}x+y\le 20\\ 0+0\le 20\\ 0\le 20\end{array}$

As, the condition obtained is $0\le 20$, which is true. Therefore, to draw the graph of the inequality $x+y\le 20$, shade the region towards the point $(0,0)$.

The graph of the given inequality $x+y\le 20$is:

Step 3. Draw the graph of the inequality 7x+10y≥90.

Convert the inequality $7x+10y\ge 90$ into equality that is convert the inequality into an equation.

Therefore, it is obtained that: $7x+10y=90$

Therefore, the equation of the boundary is $7x+10y=90$. The inequality role="math" localid="1648201648300" $7x+10y\ge 90$ has equal to sign, therefore the boundary is included in the solution. Therefore, the boundary is denoted by solid lines.

Draw the graph of the boundary line $7x+10y=90$.

Substitute 0 for $x$and find the value of $y$.

$\begin{array}{r}7\left(0\right)+10y=90\\ 10y=90\\ y=9\end{array}$

Therefore, one of the points is $(0,9)$

Substitute 0 for $y$and find the value of $x$.

$\begin{array}{r}7x+10\left(0\right)=90\\ 7x=90\\ x=\frac{90}{7}\\ x=12.86\end{array}$

Therefore the other point is $(12.86,0)$

Therefore, draw the graph of the boundary line $7x+10y=90$ by drawing a line passing through the points $(0,9)$ and $(12.86,0)$.

Now to draw the graph of the inequality $7x+10y\ge 90$, take any point which is not on the line $7x+10y=90$and substitute it in the inequality $7x+10y\ge 90$. If the condition obtained is true, then shade the region towards that point and if the condition obtained is false, then shade the region away from the point.

Let the point be role="math" localid="1648201898993" $(0,0)$and the point $(0,0)$is not on the line .

Substitute the point in the inequality $7x+10y\ge 90$.

$\begin{array}{r}7\left(0\right)+10\left(0\right)\ge 90\\ 0+0\ge 90\\ 0\ge 90\end{array}$

As, the condition obtained is $0\ge 90$, which is false. Therefore, to draw the graph of the inequality $7x+10y\ge 90$, shade the region away from the point $(0,0)$.

The graph of the given inequality $7x+10y\ge 90$ is:

Step 4. Draw the graph of the inequalities x+y≤20 and 7x+10y≥90 in one graph.

The graph of the inequalities $x+y\le 20$and $7x+10y\ge 90$in one graph is:

As, $x$ and $y$ are the number of hours. Therefore, they cannot be negative, That implies, $x\ge 0$ and $y\ge 0$. That implies the solution lies in the first quadrant.

Shade the common region of both the inequalities $x+y\le 20$and $7x+10y\ge 90$in the first quadrant to find the solution of the given system of inequalities.

Step 5. Draw the graph of the two inequalities that Kishi can use to determine how many hours she needs to work at each job if she wants to earn at least $90 per week.

The two inequalities are:

$x+y\le 20...\left(1\right)$

$7x+10y\ge 90.....\left(2\right)$

The graph of the two inequalities is: