Exercise 35 Page 261

For this exercise we can write and graph a system of inequalities. The system will have three inequalities.

• Inequality (I) represents the total amount spent on savings and housing.
• Inequality (II) represents the amount spent on savings.
• Inequality (III) represents the amount spent on housing.

For all three inequalities, we will let $x$ represent the amount spent on savings and $y$ represent the amount spent on housing. It is given that the total amount earned in one paycheck is $2000$ dollars.

Writing the Inequalities

Inequality (I): It is given that we plan to spend less than half of one paycheck on savings and checking. Thus, we can write an inequality to represent this situation as the following way. Inequality (II): It is given that we want to spend at least $10\%$ of the paycheck on savings. The phrase at least can be interpreted as greater than or equal to and represented with $\ge .$ Also, we can express $10\%$ as $0.10.$ Then, we have the following inequality. Inequality (III): It is given that we want to spend at most $30\%$ of the paycheck on housing. The phrase at most can be interpreted as less than or equal to and be represented with $\le .$ Also, we can express $30\%$ as $0.30.$ Then we can write the following equality. With these inequalities we can write the following system.

Solving the System

To determine the amount we can spend on savings and housing we can graph the system created above. The solution to the system will be the intersection of the solution sets of all three inequalities. Let's graph each inequality on its own first.

Graphing Inequality (I)

To graph the first inequality we need to first graph the boundary line. We need to write the inequality as though it were a line in slope-intercept form. Note, because the symbol is $<$ our boundary line should be dashed instead of solid. Now we need to decide which side of the line should be shaded. We can substitute any arbitrary point into the inequality to check if it is a solution. For simplicity, let's use $(0,0).$ Because $(0,0)$ is a solution to the inequality, we should shade the side of the line containing the point. Also, we should restrict the domain and range to reflect the fact that we cannot spend a negative amount of money on either expenditure.

Graphing Inequality (II)

To graph the second inequality we will follow a similar process as we did with the first inequality. First, we need to graph the boundary line. Note, because the symbol is $\ge$ our boundary line should be solid.

Now we need to decide which side of the line should be shaded. We need to include all points with $x\text{-}$values greater than or equal to $200.$ This means we should shade to the right of the line. Remember, we need to restrict the domain and range to reflect that we cannot have any negative spending amounts.

Graphing Inequality (III)

To graph the third and final inequality, we will once again begin by graphing the boundary line. Note, because the symbol is $\le$ our boundary line should be solid.

Now we need to decide which side of the line should be shaded. We need to include all points with $y$ values less than or equal to $600.$ This means we should shade below the line. Remember, we need to restrict the domain and range to reflect that we cannot have any negative spending amounts.

Combining the Solution Sets

If we show all of these individual solution sets on the same coordinate plane, we have the following graph.

Finally, we can cut away any extra information so that we view only the solution set for the system as a whole.

Any point contained in the above shaded area will be a solution for how much we can spend on savings and housing without going over our $\$1000$ budget. Let's write some example solutions.