In the coordinate geometry plane, region P is defined by all the point

Breaking Down the Info:

Note that \(a\) is standing in for \(x\), and that \(b\) is the y-coordinate.

Statement 1 Alone:

We have \(a = 4b + 8\). Plug \(b\) for \(y\) and \(4b + 8\) for \(x\) in region P, we would have :

\(3b + 12 > 2(4b + 8)\) and \(5b < -4\), which is \(b < -\frac{4}{5}\). Thus this tells us this line would be within region P if we have \(b < -\frac{4}{5}\). Since that is not guaranteed, statement 1 alone is insufficient.

Statement 2 Alone:

We can rewrite region P as \(3y - 2x > -12\). With \(b < 0\) and \(a > 3\), we can write \(3b - 2a < -6\), so with the given conditions of a and b, \(3b - 2a\) could be more than -12 or less than -12. Then statement 2 alone is insufficient.

Both Statements Combined:

We have \(a > 3\), plug in \(a = 4b + 8\) to find \(4b + 8 > 3\) and \(b> -\frac{5}{4}\). Thus the possible region of b is \(-\frac{5}{4} < b < 0\). Recall we need \(b < -\frac{4}{5}\) for the point (a, b) to be within the region, so the value of b could be less than \( -\frac{4}{5}\) or more than \(-\frac{4}{5}\). Then combined it is still insufficient.

Answer: E