The required system of inequalities is:
-2x + 3y > 15
y + 3x ≥ -4
y + 11x ≥ 6
In the given graph:
The dotted line represents the equation:
-2x + 3y = 15 → line 1.
The line towards the left of the shaded region represents the equation:
y + 3x = -4 → line 2.
The line towards the right of the shaded region represents the equation:
y + 11x = 6 → line 3.
Let's take a point in the shaded region and use the equations of the line to determine the inequality.
Consider the point (-2, 6) and substitute these values in the equations of all three lines.
Line 1:
LHS of line 1 = -2x + 3y = -2(-2) + 3(6) = 4 + 18 = 22
RHS of line 2 = 15.
Therefore, LHS > RHS.
⇒The required inequality is -2x + 3y > 15. It is a strict inequality since the equation of the line is not included in the shaded region.
Line 2:
LHS of line 2 = y + 3x = 6 +3(-2) = 6 - 6 = 0
RHS of line 2 = -4
Therefore, LHS > RHS.
⇒The required inequality is y + 3x ≥ -4. The "Equal to" symbol is used because the equation of the line is included in the shaded region.
Line 3:
LHS of line 3 = y + 11x = 6 + 11(-2) = 6 - 22 = -16
RHS of line 3 = 6
Therefore, LHS < RHS.
⇒The required inequality is y + 11x ≥ 6. The "Equal to" symbol is used because the equation of the line is included in the shaded region.
Therefore, the required system of inequalities is:
-2x + 3y > 15
y + 3x ≥ -4
y + 11x ≥ 6
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