A shepherd wants to build a rectangular sheep barn, and the graph represents the relation between the dimensions of the needed barn, where 𝑥 represents the width and 𝑦 represents the length. State the system of inequalities that describes the dimension of the barn.
And then we’re given a list of five different systems of inequalities. Before we look at those, let’s think about what we can see in the graph. On this coordinate grid, we’re only given positive 𝑥- and 𝑦-values. Since we have positive 𝑥-values and positive 𝑦-values, we can tell that we’re looking at the first quadrant in a coordinate grid, which makes sense to us because we wouldn’t have a negative length or a negative width. So we’re saying 𝑦-values greater than or equal to zero and 𝑥-values greater than or equal to zero. All five of these systems of equations list 𝑥 is greater than or equal to zero, 𝑦 is greater than or equal to zero. And so we need to consider the two other constraints.
Just above 60, we have a solid line. And all of the space above that line is shaded. This means the length of the barn must be greater than or equal to 61 meters. By the solid line, we know it can be equal to 61, and the shading above means greater than. To write this as an inequality, we say 𝑦 is greater than or equal to 61. And that is only happening in the systems for (A) and (B), which means we can eliminate options (C) through (E).
Our second constraint has a dotted line and shading underneath that dotted line. The dotted line means not equal to, and the shading below means less than, which means we need to consider these two final inequalities. Is our pink area represented by two times 𝑥 plus one is less than 177? Or 𝑥 plus 𝑦 is greater than 177? There are two things we should see here. The first is that 𝑥 plus 𝑦 is greater than 177 would have a shading above this line. In fact, we could rewrite it as 𝑦 is greater than 177 minus 𝑥. If we do that, we see that this equation would have to have a 𝑦-intercept of 177, which is not what we see in our pink shading.
On the other hand, if we divide the top equation by two on both sides, we see that 𝑥 plus 𝑦 is less than 88.5 or 𝑦 is less than 88.5 minus 𝑥, which has a 𝑦-intercept of 88.5 and a shade below that line. This means that two times 𝑥 plus 𝑦 is less than 177 most accurately reflects the information we have on this graph. So we eliminate option (B) and say that the system of inequalities we’re looking for is (A): 𝑥 is greater than or equal to zero, 𝑦 is greater than or equal to zero, 𝑦 is greater than or equal to 61, and two times 𝑥 plus 𝑦 is less than 177.