Graphical Inequalities

Firstly, rearrange this equation to make $y$ the subject by subtracting $x$ from both sides to get

 

$2y leq 8 – x$

 

Then, divide both sides by 2 to get

 

$y leq -dfrac{x}{2} + 4$

 

Drawing this is an equation, the graph would be a solid line with gradient $-frac{1}{2}$ and y-intercept 4. Once drawn, we should shade and mark the region below the line. The result looks like:

 

We’re going to treat the inequalities as equations and plot them as straight lines. The first one will be the solid plot of the line $y=1$, the second will be a solid plot of the line $y=3$, the third will be a dashed plot of the line $x=0$, and the fourth will be a dashed plot of the line $y=x$.

 

Now, we want to shade the area that is below the line $y=1$, above the line $y=3$, to the right of the line $x=0$, and above the line $y=x$.

 

The resulting graph looks like:

 

We’re going to treat the inequalities as equations and plot them as straight lines. The first one will be the solid plot of the line $y=x+1$, the second will be a solid plot of the line $x=4$, and the third will be a dashed plot of the line $y=3$

 

Now, we want to shade the area that is below the line $y=x+1$, above the line $y=-3$, and to the left of the line $x=4$

 

The resulting graph looks like:

 

Firstly, determine the equations of the $3$ lines and then from them, find the inequalities.

 

The horizontal line is clearly $y = -2$.

 

The dashed line has its y-intercept at $2$ and a gradient of $2$, so it is $y=2x + 2$

 

The final line has its y-intercept at $5$ and a gradient of $-3$, so it is $y = -3x + 5$

 

Now, the shaded area is above $y=-2$ and the line is solid, so the inequality is

 

$ygeq -2$

 

The shaded area is below $y=2x+2$ and the line is dashed, so the inequality is

 

$y <2x+2$

 

The shaded area is below $y=-3x+5$ and the line is solid, so the inequality is

 

$y leq -3x+5$

 

Therefore, the shaded area is described by the $3$ inequalities

 

$ygeq -2,hspace{2mm} y<2x+2,text{ and }yleq-3x+5$

Firstly, determine the equations of the $3$ lines and then from them, find the inequalities.

 

The vertical line is clearly $x = 2$

 

The dashed line has its y-intercept at $2$ and a gradient of $dfrac{1}{4}$, so it is $y=dfrac{1}{4}x + 2$

 

The final line has its y-intercept at $-6$ and a gradient of $2$, so it is $y = 2x -6$

 

Therefore, the shaded area is described by the $3$ inequalities:

 

$x> 2,hspace{2mm} y<dfrac{1}{4}x + 2,text{ and }ygeq-2x -6$