Question

How do you graph `x-2y<=5`?

Answer 1

`y>=1/2x-5/2`

This is more a focus on manipulating an inequality rather than just answering the question.

Explanation:

`color(blue)("Manipulating of the inequality")`

Manipulate this the same way you would an equation.

Add `color(red)(2y)` to both sides (changes it from negative to positive.

`color(green)(x-2y<=5 color(white)("dddd") ->color(white)("dddd") xcolor(white)("d") ubrace(-2ycolor(red)(+2y))<=5 color(red)(+2y)`
`color(white)("dddddddddddddddddddddddd") darr`
`color(green)(color(white)("dddddddd.ddd.") ->color(white)("dddd")xcolor(white)("..d") +0 color(white)("ddd") <= 5+2y)`

Shortcut method: move it to the other side and change the sign from subtract to add

Subtract 5 from both sides

`color(green)(color(white)("dddddddddddddd")->color(white)("dddd") x-5 <=2y)`

Divide both sides by `color(red)(2)`

`color(green)(x-5 <=2y color(white)("dddddd") ->color(white)("dddd")1/color(red)(2) x-5/color(red)(2) <= 2/color(red)(2) y`

`color(green)(color(white)("ddddddddddddddd")->color(white)("dddd")1/2x-5/2 <=y)`

Shortcut method: move the 2 from `2y` to the other side and change it from multiply to divide (everything)

Writing this in line with convention:

`y>=1/2x-5/2`

Think of the > as a birds beak. Notice that the wide part of the beak faces the y no matter which side of the inequality it is.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Plotting the inequality")`

`ul("As the symbol is ">=" the the line plotted is solid.")`

Note: If the symbol was `>` the line would be dotted

Gust so that you 'get the line' think of the inequality as
`y=1/2x-5/2`

Set `y=0=1/2x-5/2color(white)("dddd") =>color(white)("dddd") x=5`

`color(brown)("So the x-intercept is 5")`

Set `x=0 ->y=0-5/2`

`color(brown)("So the y-intercept is "-5/2)`

The inequality states `y>=" something"` so the feasible region is all the area above and on the line. (y is greater than or equal to)

`color(brown)("feasible region is all the area above and on the line")`