Question

How do you graph a linear inequality in only one variable?

Answer 1

Hint: We first take an arbitrary inequation as $x<-0.5$. Then we try to take points which have x coordinates less than $-0.5$. There is no restriction on the y coordinates. Based on the points we try to find the space or region in the 2-D plane which satisfies $x<-0.5$.

Complete step-by-step solution:
We try to take an arbitrary linear inequality in only one variable as $x<-0.5$.
The inequation $x<-0.5$ represents the space or region in 2-D plane where the x coordinates of points are valued less than $-0.5$.
We first take some points for the x coordinates where $x<-0.5$.
The values will be $x=-1,-2,-5,-10$.
We can take the y coordinate anything we want. The inequation is only based on the values of the x coordinates.
So, for y coordinate we take $y=-2,5,0,-10$ respectively for the x coordinates $x=-1,-2,-5,-10$
The points are $(x,y)=(-1,-2);(-2,5);(-5,0);(-10,-10)$.
We put these points in the graph to get

Based on the points we can measure the region.
Now we try to draw the line $x=-0.5=-{\displaystyle \frac{1}{2}}$.
Then we take all the x coordinates which have less value than $x=-{\displaystyle \frac{1}{2}}$.

The dotted line is the equation $x=-{\displaystyle \frac{1}{2}}$. All the points on the left side have x coordinates valued less than $x=-{\displaystyle \frac{1}{2}}$. So, all those points on the marked area are solutions to the inequation $x<-0.5$.

Note: We can also express the inequality as the interval system where $x<-0.5$ defines that $x\in (-\mathrm{\infty },-0.5)$. The interval for the y coordinates will be anything which can be defined as $y\in (-\mathrm{\infty },\mathrm{\infty })$. We also need to remember that the points on the line $x=-0.5$ will not be the solution for the inequation.