Answer:
- 2x -y ≥ -5
- 3x +y < 10
- x +2y > 0
Explanation:
You want a system of inequalities that will define the area within the triangle with vertices (-2, 1), (1, 7), and (4, -2), including only the left side.
Linear equations
An equation through points (x1, y1) and (x2, y2) can be written as ...
(y2 -y1)x -(x2 -x1)y = x1·y2 -x2·y1
Removing any common factor from the coefficients, and making sure the leading coefficient is positive will put this in standard form.
For points (-2, 1) and (1, 7), the equation of the line can be ...
(7 -1)x -(1 -(-2))y = (-2)(7) -(1)(1)
6x -3y = -15 . . . simplify
2x -y = -5 . . . . . divide by 3 (shading is right of the solid line)
For points (1, 7) and (4, -2), the equation of the line can be ...
(-2 -7)x -(4 -1)y = (1)(-2) -(7)(4)
-9x -3y = -30 . . . simplify
3x +y = 10 . . . . . . divide by -3 (shading is left of the dashed line)
For points (4, -2) and (-2, 1), the equation of the line can be ...
(1 -(-2))x -(-2 -4)y = (4)(1) -(-2)(-2)
3x +6y = 0 . . . simplify
x +2y = 0 . . . . divide by 3 (shading is right of the dashed line)
Inequalities
In each of the above equations, the x-coefficient is positive. This means the equal sign can be replaced with > when shading is right of the line, and with < when shading is left of the line.
There will be an "or equal to" symbol when the line is solid.
This means our inequalities are ...
- 2x -y ≥ -5
- 3x +y < 10
- x +2y > 0