Question

Graph the system of inequalities.

(x-y< or = to 1
(x+2y < 4

Answer 1

`y\ge \:-1+x,\:y<(4-x)/(2)\quad :\quad \begin{bmatrix}y\ge \:-1+x\\ y<(4-x)/(2)\end{bmatrix}\quad \mathrm{Unbounded}`

The graph is attached below.

Explanation:

Given the system of inequalities.

`\begin{bmatrix}x-y\le \:1\\ x+2y<4\end{bmatrix}`

Isolate y for
`x-y\le \:1`

`x-y\le \:1`

subtract x from both sides

`x-y-x\le \:1-x`

simplify

`-y\le \:1-x`

Multiply both sides by -1 (reverse the inequality)

`\left(-y\right)\left(-1\right)\ge \:1\cdot \left(-1\right)-x\left(-1\right)`

simplify

`y\ge \:-1+x`

now solving

`x+2y < 4`

isolate y for
`x+2y < 4`

`x+2y < 4`

subtract x from both sides

`x+2y-x<4-x`

simplify

`2y<4-x`

Divide both sides by 2

`(2y)/(2)<(4)/(2)-(x)/(2)`

Simplify

`y<(4-x)/(2)`

Graphing Method:

1. Graph each inequality separately

2. Choose a test point to determine which side of the line needs to be shaded

3. The solution to the system will be the area where the shadings from each inequality overlap one another.

Thus,

`y\ge \:-1+x,\:y<(4-x)/(2)\quad :\quad \begin{bmatrix}y\ge \:-1+x\\ y<(4-x)/(2)\end{bmatrix}\quad \mathrm{Unbounded}`

The graph is attached below.