we have
\(y < 5x + 2\) -------> inequality \(1\)\(y \geq \frac{1}{2}x + 1\) -------> inequality \(2\)
using a graph tool
see the the attached figure N \(1\)
the solution of the system is the shaded area
\(case 1) (-3, 2)\)
This point satisfies the inequality \(2\) but does not satisfy the inequality \(1\).
therefore
it is not a solution of the system
see the attached figure N \(2\)
inequality \(1\)
\((-3,2)\\ x=-3\\ y=2\)
\(2 < 5*(-3) + 2\)
\(2 < -13\) -----> is not true
inequality \(2\)
\((-3,2)\\ x=-3\\ y=2\)
\(2 \geq \frac{1}{2}*(-3) + 1\)
\(2 \geq -\frac{1}{2}\) -----> is true
\(case 2) (-1, 3)\)
This point satisfies the inequality \(2\) but does not satisfy the inequality \(1\).
therefore
it is not a solution of the system
see the attached figure N \(2\)
inequality \(1\)
\((-1, 3)\\ x=-1\\ y=3\)
\(3 < 5*(-1) + 2\)
\(3 < -3\) -----> is not true
inequality \(2\)
\((-1, 3)\\ x=-1\\ y=3\)
\(3 \geq \frac{1}{2}*(-1) + 1\)
\(3 \geq \frac{1}{2}\) -----> is true
\(case 3) (0, 2)\)
This point satisfies the inequality \(2\) but does not satisfy the inequality \(1\).
therefore
it is not a solution of the system
see the attached figure N \(2\)
inequality \(1\)
\((0, 2)\\ x=0\\ y=2\)
\(2 < 5*(0) + 2\)
\(2 < 2\) -----> is not true
inequality \(2\)
\((0, 2)\\ x=0\\ y=2\)
\(2 \geq \frac{1}{2}*(0) + 1\)
\(2 \geq 1\) -----> is true
\(case 4) (1, 2)\)
This point satisfies the inequality \(2\) and satisfies the inequality \(1\).
therefore
it is a solution of the system
see the attached figure N \(2\)
inequality \(1\)
\((1, 2)\\ x=1\\ y=2\)
\(2 < 5*(1) + 2\)
\(2 < 7\) -----> is true
inequality \(2\)
\((1, 2)\\ x=1\\ y=2\)
\(2 \geq \frac{1}{2}*(1) + 1\)
\(2 \geq \frac{3}{2}\) -----> is true
\(case 5) (2,-1)\)
This point satisfies the inequality \(1\) but does not satisfy the inequality \(2\).
therefore
it not a solution of the system
see the attached figure N \(2\)
inequality \(1\)
\((2, -1)\\ x=2\\ y=-1\)
\(-1 < 5*(2) + 2\)
\(-1 < 12\) -----> is true
inequality \(2\)
\((2, -1)\\ x=2\\ y=-1\)
\(-1 \geq \frac{1}{2}*(2) + 1\)
\(-1 \geq 2\) -----> is not true
\(case 6) (2, 2)\)
This point satisfies the inequality \(1\) and satisfies the inequality \(2\).
therefore
it is a solution of the system
see the attached figure N \(2\)
inequality \(1\)
\((2, 2)\\ x=2\\ y=2\)
\(2 < 5*(2) + 2\)
\(2 < 12\) -----> is true
inequality \(2\)
\((2, 2)\\ x=2\\ y=2\)
\(2 \geq \frac{1}{2}*(2) + 1\)
\(2 \geq 2\) -----> is true
therefore
the answer is
\(case 4) (1, 2)\\ case 6) (2, 2)\)
