# Inequations with two variables

## Problems

Solve the following inequations and give the region in the plane where they are satisfied:

1. $y-3x > 2$

2. $2(x-y)-3(y+2) < 2x+1$

3. $\dfrac{-x+4y}{3}-x \geqslant 2y+x$
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### Development:

We are going to solve three inequations. We will give the expression of the inequation as $y < ax+b$ and will say what points of the plane we take.

1. $y-3x > 2 \Rightarrow y > 3x -2$. The solution region is above the straight line (we have an inequality of the type $>$).

2. $2(x-y)-3(y+2) < 2x+1 \Rightarrow 2x-2y-3y-6 < 2x+1 \Rightarrow$ $\Rightarrow 2x-2x-6-1 < 5y \Rightarrow y > -\dfrac{7}{5}$. The solution region is above the straight line.

3. $\dfrac{-x+4y}{3}-x\geqslant 2y+x \Rightarrow \dfrac{-x+4y-3x}{3} \geqslant 2y+x \Rightarrow$ $\Rightarrow -x+4y-3x \geqslant 6y+3x \Rightarrow -x-3x-3x \geqslant 6y-4y \Rightarrow y \leqslant \dfrac{-7}{2}x$. The solution region is below the straight line, taking also the line points.

### Solution:

1. $y > 3x -2 \Rightarrow$ Points above de line $y-3x = 2$ without the points in the line.

2. $y > -\dfrac{7}{5} \Rightarrow$ Points above de line $y = -\dfrac{7}{5}$ without the points in the line.

3. $y \leqslant \dfrac{-7}{2}x \Rightarrow$ Points below de line $y = \dfrac{-7}{2}x$ with the points in the line.
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