Question Video: Graphs of Linear Inequalities

Which of the figures is the correct graph of 𝑦 < 𝑥? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

02:11

Video Transcript

Which of the following figures is the correct graph of 𝑦 is less than 𝑥?

We notice that all five of the figures contain the line 𝑦 is equal to 𝑥. This straight line passes through the origin zero, zero and has a slope or gradient of positive one. When dealing with inequalities as in this case, we’ll need to shade the correct area. This immediately rules out option (C) as this simply represents the line 𝑦 is equal to 𝑥.

In options (B) and (D), the line is broken or dotted, whereas in options (A) and (E), the line is solid. When the line is broken, the equation will be strictly greater than or strictly less than. This is true for this question, and figure (B) represents the values where 𝑦 is less than 𝑥. This is because the region shaded is below the line. On the other hand, figure (D) represents the values where 𝑦 is greater than 𝑥 as the region shaded is above the line. Below the line, any 𝑦-value will be less than the paired 𝑥-value. This means that the correct answer is figure (B).

For completeness, figure (A) represents the inequality 𝑦 is less than or equal to 𝑥. And figure (E) represents the inequality 𝑦 is greater than or equal to 𝑥.

We could check this answer by selecting any coordinate in the shaded region. In figure (B), we see that the point with coordinates four, two lies in the shaded region. The 𝑦-coordinate is equal to two, and the 𝑥-coordinate is equal to four. Therefore, this satisfies 𝑦 is less than 𝑥. The same is true for any other points we can pick in figure (B).

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