Question

Examine the following graph of the system of inequalities y≤x2−4x−3 and y<−2x+4. A is the area below the line and the parabola. B is the area below the line but above the parabola. C is the area above the line and the parabola. D is the area below the parabola but above the line.© 2018 StrongMind. Created using GeoGebra. Which section of the graph represents the solution set to the system of inequalities?

Answer 1

The solution set to a system of inequalities represents the area that contains points that satisfy both inequalities.

The best way to answer the question is to choose one point from each area and check if they satisfy both.

Let's start by selecting a point in area A. Let's use (-6, 0).

`\begin{gathered} 0\leq(-6)^2-4(-6)-3 \\ 0\leq36+24-3 \\ 0\leq57\text{ TRUE} \\ \\ 0<-2(-6)+4 \\ 0<12+4 \\ 0<16\text{ TRUE} \end{gathered}`

Because out test point (-6, 0) satisfies both inequalities, then the entire area that contains it is the solution. We no longer have to test other points.

The answer is A.

The solution set is also the intersection of the graphs of the two inequalities. So you may refer to the shaded regions and you'll see that area A is shaded red and blue at the same time.