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.