Question Video: Identifying Regions That Represent the Solutions to a System of Inequalities Mathematics

Video Transcript

Which region on the graph contains solutions to the set of inequalities 𝑦 is greater than 𝑥 and 𝑦 is less than or equal to two 𝑥 minus four?

The graph we are given has been split into seven regions, and we need to identify which of the region’s satisfies our two inequalities.

Let’s begin by considering the inequality 𝑦 is greater than 𝑥. We know that the line 𝑦 equals 𝑥 will pass through the origin. And every point that lies on this line will have the same 𝑥- and 𝑦-coordinate, for example, the point five, five and the point negative five, negative five. As 𝑦 must be greater than 𝑥, our inequality is strict, and this is represented by a dashed line on our graph.

The region that satisfies this inequality is above the line 𝑦 equals 𝑥. For example, the point zero, five has a 𝑦-coordinate greater than its 𝑥-coordinate. We can therefore rule out regions C, D, E, and F.

Let’s now consider our second inequality and firstly the line with equation 𝑦 is equal to two 𝑥 minus four. This equation is written in the form 𝑦 is equal to 𝑚𝑥 plus 𝑏. And the line will therefore have a slope, or gradient, equal to two and a 𝑦-intercept of negative four.

Since the inequality is less than or equal to, we will draw a solid line. The region we require this time is below the line, and this rules out regions A and B. We can therefore conclude that the region on the graph that satisfies the inequalities 𝑦 is greater than 𝑥 and 𝑦 is less than or equal to two 𝑥 minus four is region G.

We can check this by selecting a point in this region, for example, the point seven, eight. This has an 𝑥-coordinate equal to seven and a 𝑦-coordinate equal to eight. The first inequality is therefore satisfied, as eight is indeed greater than seven. Substituting the values of 𝑥 and 𝑦 into the second inequality, we have eight is less than or equal to two multiplied by seven minus four. The right-hand side is equal to 10, so we have eight is less than or equal to 10. As this statement is correct, the point also satisfies the second inequality, and this confirms that the correct answer is region G.