Question

Write the inequality shown by the shaded region in the graph with the boundary line 2x + 2y = -6

Answer 1

The Solution:

Given the equation of a line below:

`2x+2y=-6`

Step 1:

We shall determine the x-intercept and y-intercept of the given line.

x-intercept: The value of x when y=0

`\begin{gathered} \text{When y=0} \\ 2x+2(0)=-6 \\ 2x=-6 \end{gathered}`

Dividing both sides by 2, we get

`\begin{gathered} x=(-6)/(2)=-3 \\ \text{ So,} \\ \text{ The x-intercept = (-3,0)} \end{gathered}`

Similarly,

y-intercept: the value of y when x=0

`\begin{gathered} 2(0)+2y=-6 \\ \\ 2y=-6 \end{gathered}`

Dividing both sides by 2, we get

`\begin{gathered} y=(-6)/(2)=-3 \\ \text{ hence,} \\ \text{ The y-intercept = (0,-3)} \end{gathered}`

Determine the inequality symbol that will be used to replace the equality sign.

If the straight line is unbroken, it means the points on the line are inclusive. So, the inequality symbol will not be a strict inequality. It will be one of these two inequalities:

`\leq\text{ or }\ge`

Determine the exact inequality symbol that will represent the shaded region.

If the shaded region is below the line, the correct inequality will be:

`\leq`

But where the shaded region is above the line, the correct inequality symbol will be:

`\ge`

From the given graph, it is clear that the shaded region is above the line. So, it follows that the correct inequality is:

`\ge`

Therefore, the correct answer is:

`2x+2y\ge-6`