The solutions of a linear equation form a line in a coordinate plane. Linear inequalities, on the other hand, are sets of coordinates that create an entire region of a coordinate plane. This begs the question Why does the graph of an inequality contain a region? Consider the following inequality. The boundary line to the inequality is $y=x.$ It's the line that passes through all points where $x$ and $y$ have the same value. These include $(\text{-}1,\text{-}1),$ $(0,0),$ $(1,1),$ etc. The inequality $y\le x$ describes all the points where the $y$-coordinate is less than or equal to the $x$-value. For $x=4,$ the inequality becomes Thus, for all points with $x=4,$ if the corresponding $y$-value is less than or equal to $4,$ the point is a solution to the inequality.

The reasoning can be applied to several $x$-values. Applying it to all $x$-values creates the entire region below the line $y=x.$

This means the area on and below the line $y=x$ contains all pairs of $x$ and $y$ that satisfy the inequality $y\le x.$ Thus, an entire region is created.