Given below are the graphs of two lines, \(y=-0.5 x + 5\ \) and \(y=-1.25 x + 8,\) and several regions and points are shown. Note that \(C\) is the region that appears completely white in the graph.
The blue line has equation \(y=−1.25x+8\) and the red line has equation \(y=−0.5x+5.\)
Many answers are possible. We give one for each region.
The point \((0,0)\) lies in region A: Substituting this point into the inequalities we have \begin{eqnarray*} 0&\leq&-0.5\cdot 0 + 5=5\\ 0&\leq&-1.25\cdot 0 + 8=8, \end{eqnarray*} which is true.
The point \((0,6)\) lies in region B: Substituting this point into the inequalities we have \begin{eqnarray*} 6&\geq&-0.5\cdot 0 + 5 = 5\\ 6&\leq&-1.25\cdot 0 + 8 = 8, \end{eqnarray*} which is true.
The point \((8,10)\) lies in region C: Substituting this point into the inequalities we have \begin{eqnarray*} 10&\geq&-0.5\cdot 8 + 5=1\\ 10&\geq&-1.25\cdot 8 + 8=-2, \end{eqnarray*} which is true.
The point \((10,-1)\) lies in region D: Substituting this point into the inequalities we have \begin{eqnarray*} -1&\leq&-0.5\cdot 10 + 5=0\\ -1&\geq&-1.25\cdot 10 + 8=-4.5, \end{eqnarray*} which is true.
We first observe that every point in the 3rd quadrant has negative \(x\)- and \(y\)-coordinates. So we have to show that for any point with negative \(x\) and negative \(y\) coordinates the two inequalities \begin{eqnarray*} y&\leq&-0.5 x + 5\\ y&\leq&-1.25 x + 8 \end{eqnarray*} are satisfied.
Note that for any negative value of \(y\), the left hand side of both inequalities will be negative. Similarly, for any negative value of \(x\), the right hand side of both inequalities will be positive. It is true that any negative number is smaller than any positive number. Therefore, both inequalities are satisfied for negative values of \(x\) and \(y\).
Commentary
The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?
The last part of this problem requires the students to make a general argument without using specific numbers (SMP 4) and instead to recognize the structure of the inequalities (SMP 7).
The task could be used in many instructional settings, but having students share their thinking and respond to each others' arguments would provide a rich learning experience.