Question

Which system of inequalities represents the graph?

Answer 1

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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