Question

andrew is working two summer jobs, making $14 per hour lifeguarding and making$10 per hour washing cars. in a given week, he can work at most 15 total hours and must earn no less than $170. also, he must work at least 9 hours lifeguarding and at most 3 hours washing cars. if x represents the number of hours lifeguarding and y represents the number of hours washing cars, write and solve a system of inequalities qraphically and determine one possible solution.number of inequalities can change.

Answer 1

Here, we want to write inequalities

We need to know that the term at most in inequality means less than or equal to

The term at least mean greater than or equal to

From the question, we have some defined variables

x represents the number of hours lifeguarding and y represents the number of hours washing cars

In a week, the highest number of hours he can work is 15

Thus, if we add both, the result cannot be more than 15 hours

Mathematically, this is;

`x\text{ + y}\leq\text{ 15}`

Secondly, the lowest earnings he can have per week is $170

Earnings of $10 per hour for y hours will give 10 * y = $10y

Earnings of $14 per hour for x hours will give 14 * x = $14x

The addition of both gives at least $170

Mathematically, this will be;

`14x\text{ + 10y }\ge\text{ 170}`

We still have more inequalities;

We are told he must spend at least 9 hours lifeguarding

Mathematically, that will be;

`x\text{ }\ge\text{ 9}`

And lastly, he must spend at most 3 hours washing cars

That will be;

`y\text{ }\leq\text{3}`

So, there are 4 inequalities to write

Now, we want to plot the graphs of the inequality

Let us start with;

`x\text{ + y }\leq\text{ 15}`

The general form of the of a straight line;

`y\text{ = mx + c}`

where m represents the slope and c represents the intercept on the y-axis

To plot, we remove the inequality and write the equation in its normal form as;

`\begin{gathered} x\text{ + y =15} \\ \\ In\text{ general form;} \\ \\ y\text{ = -x + 15} \end{gathered}`

To make a plot, we need the x-intercept and the y-intercept

The y intercept is at y = 15

To get the x intercept, we set y to 0; thus we have;

`\begin{gathered} 0\text{ = -x + 15} \\ \\ x\text{ = 15} \end{gathered}`

So this mean that the x and y intercepts are 15

To polt this, we mark the point 15 on the x-axis and also mark the point 15 on the y-axis

Then we draw a line joining these two points

In the correct inequality from, we will need to shade the points that are below the graph since it is an inequality and it is quite different from the normal form

For the second inequaity, we do it as we have done above

We start by writing the inequality in the general graph form after replacing the inequality sign with an equality sign

We have this as follows;

`\begin{gathered} 14x\text{ + 10y = 170} \\ \\ 10y\text{ = -14x + 170} \\ \\ \text{divide through by 10;} \\ \\ y\text{ = -1.4x + 17} \end{gathered}`

The y-axis from above is 17

To get the x-intercept, we simply proceed to equate y to 0

`\begin{gathered} 0\text{ = -1.4x + 17} \\ \\ 1.4x\text{ = 17} \\ \\ x\text{ = }\frac{17}{1.4\text{ }} \end{gathered}`

This is a little after 12 and before 13

So we join these two points with a straight line, and shade the parts above the line since it is a greater than or equal to graph

For;

`y\text{ }\leq\text{ 3}`

We draw a staright line that passes through y = 3 and is parallel to the x-axis

We shade the parts below this line

And lastly for ;

`x\text{ }\ge\text{ 9}`

We draw a straight line crossing the point x = 9, parallel to the y-axis ; then shade the parts on the right hand side

Now, we want to determine a possible solution

One possible solution is a point where the lines will meet

We then proceed to mark this point and trace the values to both the horizontal and the vertical axes