Which of the following figures is the correct graph of 𝑦 is less than or equal to 𝑥?
All five of the figures include the line 𝑦 is equal to 𝑥. This is a straight line that passes through the origin zero, zero and has a slope or gradient of positive one. Figure (C) just contains this line. Therefore, this represents 𝑦 is equal to 𝑥. We can therefore rule out this option.
The other four figures have shaded regions which represent inequalities. They will represent greater than, greater than or equal to, less than, and less than or equal to. When we have a broken or dotted line, our graph will be strictly greater than or strictly less than. This is true of options (B) and (D). In option (B), the shaded area is below the line 𝑦 equals 𝑥. Therefore, this represents 𝑦 is less than 𝑥. Option (D) represents 𝑦 is greater than 𝑥. As both of these are strictly greater than and strictly less than, we can rule out these two options.
We are now left with options (A) and (E). In option (A), we have shaded below the line 𝑦 equals 𝑥. Therefore, this corresponds to 𝑦 is less than or equal to 𝑥. Option (E) represents the inequality 𝑦 is greater than or equal to 𝑥. This means that we can rule out this figure, and the correct answer is figure (A).
We can check this by selecting a point in our shaded region. For example, the point with coordinates one, negative two is shaded in figure (A). Negative two, the 𝑦-coordinate, is less than one. Therefore, it does satisfy the inequality 𝑦 is less than or equal to 𝑥.