Question

The boundary line of a linear inequality passes through the points (3,0) and (0,6). The points (3,0) and (2,6) are not solutions of the inequality. Write the linear inequality

Answer 1

`y<-2x+6`

Explanation:

The boundary line of a linear inequality passes through the points (3, 0) and (0, 6).

The points (3, 0) and (2, 6) are not solutions of the inequality.

We want to find the equation of the linear inequality.

First, we will simply find the line that crosses the points (3, 0) and (0, 6).

We can determine the slope to be:

`\displaystyle (6-0)/(0-3)=(6)/(-3)=-2`

Notice that (0, 6) is the y-intercept.

Then by the slope-intercept form:

`y=mx+b`

Our equation will be:

`y=-2x+6`

Now, we will need to determine our inequality sign.

First, (3,0) is on the boundary line yet it is not a solution. Therefore, our line is dotted. That is: we do not have the "or equal to" statement.

So, our inequality signs can only be either < or >.

We are given that (2, 6) is not a solution.

So, we can test for the sign. Substitute 2 for x and 6 for y:

`\displaystyle 6\text{ }?\text{ }-2(2)+6`

Evaluate:

`\displaystyle 6\text{ } ?\text{ } 2`

Since 6 is not a solution, the above statement needs to be false.

It will be false is the sign is <, since 2 is not greater than 6.

Therefore, our sign is <.

And our inequality is:

`y<-2x+6`