Video Transcript
Which region on the graph contains solutions to the set of inequalities 𝑦 is less than two, 𝑦 is less than or equal to negative 𝑥, and 𝑥 is less than one?
To begin answering this question, we will identify the three lines on our graph that represent the equations 𝑦 is equal to two, 𝑦 is equal to negative 𝑥, and 𝑥 is equal to one. Any equation in the form 𝑦 equals 𝑎 where 𝑎 is some constant will correspond to a horizontal line. This means that the graph of 𝑦 equals two is a horizontal line that passes through two on the 𝑦-axis. In the same way, any equation of the form 𝑥 equals 𝑎 where once again 𝑎 is some constant will correspond to a vertical line. The equation 𝑥 equals one is a vertical line passing through one on the 𝑥-axis.
We notice that both of these lines are broken or dashed. This is because the inequalities are strictly less than. When our inequality sign is less than or equal to or greater than or equal to, we will have a solid line. The graph of the equation 𝑦 equals negative 𝑥 passes through the points one, negative one and two, negative two. This is because the 𝑦-coordinate must be the negative of the 𝑥-coordinate. This line will also pass through the points negative one, one and negative two, two. The line 𝑦 equals negative 𝑥 also passes through the origin and is the diagonal line shown on the graph.
We can now consider which region on the graph satisfies the three inequalities. As 𝑦 is less than two, the region must be below the line 𝑦 equals two. Likewise, we are told that 𝑥 is less than one. Therefore, the region must be to the left-hand side of the line 𝑥 equals one. Finally, as 𝑦 is less than or equal to negative 𝑥, our region must be below the line 𝑦 equals negative 𝑥. The region that satisfies all three of these inequalities is region B. We can therefore conclude that the region on the graph that contains solutions to the set of inequalities 𝑦 is less than two, 𝑦 is less than or equal to negative 𝑥, and 𝑥 is less than one is region B.
We can check this answer by selecting a point that lies in this region, for example, the point with coordinates negative two, one. Our 𝑥-coordinate here is negative two. And the 𝑦-coordinate is one. One is less than two. Therefore, the first inequality holds. The second inequality is slightly more complicated. Substituting in our coordinates, we have one is less than or equal to negative negative two. This is the same as one is less than or equal to two. As this is also true, the second inequality holds. Finally, we have negative two is less than one. The inequality 𝑥 is less than one also holds for the point negative two, one. We can therefore conclude that the point negative two, one which lies in region B satisfies all three of the inequalities.
