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Write a system of inequalities to represent the situation. One inequality should represent the total amount spent on savings and housing, one should represent the amount spent on savings, and one will represent the amount spent on housing.
Example Solution: $300 on savings and $500 on housing
For this exercise we can write and graph a system of inequalities. The system will have three inequalities.
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.
at leastcan be interpreted as greater than or equal to and represented with ≥. Also, we can express 10 % as 0.10. Then, we have the following inequality. x ≥ 0.10 * 2000 ⇒ x ≥ 200 Inequality (III): It is given that we want to spend at most 30 % of the paycheck on housing. The phrase
at mostcan be interpreted as less than or equal to and be represented with ≤. Also, we can express 30 % as 0.30. Then we can write the following equality.
y ≤ 0.30* 2000 ⇒ y ≤ 600 With these inequalities we can write the following system. x+y < 1000 x ≥ 200 y ≤ 600
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.
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. x+y=1000 ⇒ y=- x+1000 Note, because the symbol is < our boundary line should be dashed instead of solid.
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 ≥ 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-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.
To graph the third and final inequality, we will once again begin by graphing the boundary line. Note, because the symbol is ≤ 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.
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. &$200 on savings and $200 on housing. &$300 on savings and $500 on housing. &$800 on savings and $100 on housing.