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

Jun 18, 2018

$y \ge \frac{1}{2} x - \frac{5}{2}$

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

#### Explanation:

$\textcolor{b l u e}{\text{Manipulating of the inequality}}$

Manipulate this the same way you would an equation.

Add $\textcolor{red}{2 y}$ 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)
$\textcolor{w h i t e}{\text{dddddddddddddddddddddddd}} \downarrow$
$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddd.ddd.") ->color(white)("dddd")xcolor(white)("..d") +0 color(white)("ddd}} \le 5 + 2 y}$

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

Subtract 5 from both sides

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddddd")->color(white)("dddd}} x - 5 \le 2 y}$

Divide both sides by $\textcolor{red}{2}$

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

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddddddd")->color(white)("dddd}} \frac{1}{2} x - \frac{5}{2} \le y}$

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

Writing this in line with convention:

$y \ge \frac{1}{2} x - \frac{5}{2}$

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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Plotting the inequality}}$

$\underline{\text{As the symbol is ">=" the the line plotted is solid.}}$

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

Gust so that you 'get the line' think of the inequality as
$y = \frac{1}{2} x - \frac{5}{2}$

Set $y = 0 = \frac{1}{2} x - \frac{5}{2} \textcolor{w h i t e}{\text{dddd") =>color(white)("dddd}} x = 5$

$\textcolor{b r o w n}{\text{So the x-intercept is 5}}$

Set $x = 0 \to y = 0 - \frac{5}{2}$

$\textcolor{b r o w n}{\text{So the y-intercept is } - \frac{5}{2}}$

The inequality states $y \ge \text{ something}$ so the feasible region is all the area above and on the line. (y is greater than or equal to)

$\textcolor{b r o w n}{\text{feasible region is all the area above and on the line}}$