Graphing Inequalities

¶ … Inequalities According to the information provided, the graph shown includes all of the possible combinations of refrigerators and televisions that would fit into the 18 wheeler that is going to deliver them to the Burbank Buy More store. It shows not only the maximum number of each item that would fit, but also the possible combinations of each individual item. To find the maximum number of refrigerators that would fit, one looks for the number of refrigerators that could fit with zero televisions.

Likewise, to determine the maximum number of televisions that would fit, one looks for the number of televisions that would fit with zero refrigerators. Knowing these parameters gives one guidelines for the upper limit of each of the commodities. The graph has refrigerators on the x axis and televisions on the y-axis. By looking at the graph, one sees that the maximum number of refrigerators that would fit is 100. This gives a coordinate of (100,0) because when y=0, x=100. The maximum number of televisions that would fit into the space is 300.

This gives a coordinate of (0,300) because when x=0, y=300. These coordinates can then be used to help develop the equation for the linear equality established by the graph. The slope of the line is determined by looking at the rise over the run. The rise is the difference in the y coordinates and the run is the difference in the x coordinates. The rise would be 330-0. The run would be 0-110.

Therefore, the slope of the line would be determined by dividing 330 by -110, and the slope of the line is -3. The y-intercept for the line is 330, which is determined by looking at the graph. Therefore, the equation for the line would be y = -3x + 330 if it were an equality. However, the truck can, but does not have to, contain a full load. Anything combination of positive numbers on the line as well as those underneath and to the left of the line would fulfill the conditions.

This is referred to as a linear inequality. This linear inequality is a solid line not a dashed line, because the points on the line can be included in the solution set. The equation for that inequality is y ? -3x + 330. The steps in the solution are below: y=mx + b 330 = 0x + b b= 330 0= 110m + b 0=110m + 330 -330= 110m m = -3 y ? -3x + 330 One can use the graph to see if certain combinations of televisions and refrigerators can be included in the shipment.

The numbers are converted into a point, and that point is used as a test point to see if they fall within the shaded area of the graph. If the test point is in shaded area, the combination can fit. If the test point is outside of the shaded area, then the combination will not work.

This can also be determined by plugging the numbers into the equation for the linear equality, which can provide more precise answers than a quick visual assessment from a graph, particularly if the point is close to the line created by the linear inequality. The first question asked is: will the truck hold 71 refrigerators and 118 TVs? Those numbers are plugged into the equation y = ? -3x + 330. The number sentence resulting is: 118 ? -3(71) + 330? Simplified, that sentence reads 118 ? 117. Clearly that is false.

Therefore, the truck cannot hold that combination of TVs and refrigerators. Instead, if there are 71 refrigerators, the truck can hold up to 117 TVs, but not 118. Next, the text asks whether the truck will hold 51 refrigerators and 176 TVs? The resulting number sentence becomes 176 ? -3(51) + 330. Simplified, that reads 176 ? 177. This is a true statement. Therefore, the truck will hold that combination of TVs and refrigerators. The next question asks how many TVs it could buy.