Solve the following systems of inequalities graphically then give three ordered pairs satisfying the inequalities. Show that the ordered pairs satisfy the inequalities.
Given that,
$y>4x-3\phantom{\rule{0ex}{0ex}}y\ge -2x+3$
Write the given inequality in equation form,
$y=4x-3...\left(1\right)\phantom{\rule{0ex}{0ex}}y=-2x+3...\left(2\right)$
Equation (1),
When x = 0, y=-3,
When y = 0, $x=\frac{3}{4}=0.75$
Equation (1) passes through (0, -3) and (0.75, 0).
Equation (2),
When x = 0, y = 3
When y = 0, $x=\frac{3}{2}=1.5$
Equation (2) passes through (0, 3) and (1.5, 0).
Subtracting (1) from (2),
$y-y=-2x+3-4x+3\phantom{\rule{0ex}{0ex}}-6x+6=0\phantom{\rule{0ex}{0ex}}6x=6\phantom{\rule{0ex}{0ex}}x=1$
Substitute x = 1 in (1),
$y=4-3=1$
So intersection point of (1) and (2) is (1, 1).
Now take three points (1, 6), (0, 4) and (2, 7).
Substitute these three points in given inequality,
At (1, 6),
$6>4-3\Rightarrow 6>1\phantom{\rule{0ex}{0ex}}6\ge -2+3\Rightarrow 6\ge 1$
At (0, 4),
$4>0-3\Rightarrow 4>-3\phantom{\rule{0ex}{0ex}}4\ge 0+3\Rightarrow 4\ge 3$
At (2, 7),
$7>8-3\Rightarrow 7>5\phantom{\rule{0ex}{0ex}}7\ge -4+3\Rightarrow 7\ge -1$
Hence these three points (1, 6), (0, 4) and (2, 7) satisfy the given inequality.
Graph:
[Shaded region represents the solution set of given equality included (1, 6), (0, 4) and (2, 7)]