Graphs of linear systems and inequalities

An inequality describes a region of the plane rather than a line. Graph the boundary line first, then decide which side satisfies the inequality.

The boundary is solid when the inequality includes equality, as in $\leq$ or $\geq$, and dashed when it is strict, as in $<$ or $>$. That detail is tested directly.

To choose the side, test a point not on the line — $(0,0)$ whenever possible. If it satisfies the inequality, shade its side; if not, shade the other.

For a system of inequalities, the solution is where all the shaded regions overlap, so a point is a solution only if it satisfies every inequality at once. Questions often give a point and ask whether it qualifies, which you answer by substituting into each in turn.

A quick shortcut for $y$ alone on the left: $y >$ means shade above the line, $y <$ means shade below.

Is $(1, 2)$ a solution of $y \leq -x + 5$ and $y \geq 0.5x + 1$?
Check each in turn: $2 \leq 4$ is true, and $2 \geq 1.5$ is true. Both hold, so yes.