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{{ printedBook.courseTrack.name }} {{ printedBook.name }} # Solving Systems of Linear Inequalities Graphically

## Exercise 1.7 - Solution

a

Let be the number of hot dogs sold per week and let be the number of sodas sold per week. The team would like to sell at least hot dogs and sodas per week. Therefore, must be greater than or equal to and must be greater than or equal to The price of one hot dog is Therefore, the expression represents the amount earned by selling hot dogs. Similarly, since the price of one soda is the expression represents the amount earned by selling sodas. We are told the goal is earning at least We can combine the three inequalities we have written to form a system of inequalities.

b

Let the horizontal axis be the axis and the vertical axis be the axis. Let's graph the inequalities one at a time.

### Inequality (I)

To obtain the boundary line we replace the inequality sign with an equals sign. The line is a vertical line which passes through Since the inequality is not strict, the line will be solid. The inequality states that is greater than or equal to Therefore, we will shade the half-plane to the right of the line. ### Inequality (II)

The boundary line related to the second inequality is This is a horizontal line whose intercept is Since is greater than or equal to we will shade the region above the line. It will be solid because the inequality is not strict. ### Inequality (III)

To draw the third inequality, first let's isolate
Solve for
Let's consider the boundary line. Note that the line is written in slope-intercept form. To graph it, we will plot its intercept and use the slope to find another point on the line. Then, we will connect these points with a straight edge. The line will be solid because the inequality is not strict. To determine the half-plane we should shade, we will test a point. If substituting the coordinates of this point into the inequality produces a true statement, we will shade the region that contains the point. Otherwise, we will shade the opposite region. For simplicity, we will test the point
Since did not produce a true statement, we will shade the region that does not contain it. The solution to this system of inequalities is where the three shadings overlap. c

To name one possible solution, we need any point from the overlapping area. We will plot the point in the graph and name it. As we see above, one possible solution is In the context of the problem, it means that hot dogs and sodas were sold.