Question

Which system of linear inequalities is represented by thegraph?+ 3 and 3x-y> 22...4-5 -4 -3 -2 -1.O ye**+O yz}x+3 and 3x –y> 2O yz}x+3 aO yax+3 and 2x -y> 22-+ 3 and 3x + y > 2

Answer 1

Inequalities

We are given a graph where a shaded area is shown. It's bounded by a solid line.

We first need to find the equation of that line by selecting any pair of clear points on the line.

For example, I can see the points (-4,1) and (0,3).

To find the equation of the line passing through them we use the following equation.

The equation of a line passing through points (x1,y1) and (x2,y2) can be found as follows:

`\displaystyle y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1)`

Substituting the given points:

`\displaystyle y-1=(3-1)/(0+4)(x+4)`

Simplifying:

`\begin{gathered} \displaystyle y-1=(2)/(4)(x+4) \\ \displaystyle y-1=(1)/(2)(x+4) \\ 2y-2=x+4 \\ 2y=x+6 \\ y=(1)/(2)x+3 \end{gathered}`

Since the shaded area is below the line, the inequality is:

`y\leq(1)/(2)x+3`

Note the symbol is 'less or equal' and not 'less' because the line is solid, which includes equality, or the points in the line.