Video Transcript
Which region on the graph contains solutions to the set of inequalities 𝑦 is less than two, 𝑦 is greater than or equal to negative 𝑥, and 𝑥 is less than one?
In order to answer this question, we begin by considering what the graphs of the equations 𝑦 equals two, 𝑦 equals negative 𝑥, and 𝑥 equals one would look like. Any equation 𝑦 equals some constant 𝑎 will correspond to a horizontal line. The equation 𝑦 equals two will, therefore, be a horizontal line passing through two on the 𝑦-axis. We draw this as a broken line, as the inequality is strictly less than. In a similar way, any equation of the form 𝑥 equals 𝑎, where 𝑎 is some constant, will be a vertical line. The equation 𝑥 equals one corresponds to a vertical line passing through one on the 𝑥-axis. Once again, this is a broken line, as the inequality sign is strictly less than.
We know that any point that lies on the equation 𝑦 equals negative 𝑥 will have a 𝑦-coordinate equal to the negative of the 𝑥-coordinate. For example, the points two, negative two and one, negative one lie on the line 𝑦 equals negative 𝑥. The points negative one, one and negative two, two will also lie on this line. This line 𝑦 equals negative 𝑥 also passes through the origin as shown.
Now that we’ve drawn the lines of these three equations, we need to consider the region that satisfies the inequalities. Firstly, 𝑦 is less than two. Therefore, our region must lie below the line 𝑦 equals two. In a similar way, 𝑥 is less than one. Therefore, our region must lie to the left of 𝑥 equals one. Finally, we have 𝑦 is greater than or equal to negative 𝑥. This means that our region must lie above the line 𝑦 equals negative 𝑥. The region that satisfies all three of these inequalities is therefore region A.
