What if the shaded regions never overlap?

If the shaded regions don't overlap, the system has no solution. This often happens with parallel lines that are shaded in opposite directions, pointing away from each other.

Do I always have to use (0,0) as a test point?

No, you can use any point on the graph as long as it does not sit exactly on the boundary line you are testing. We usually use $(0,0)$ just because the math is the easiest to calculate!

Is a point on a dashed line part of the solution?

No. A dashed line represents a strict inequality ($ $). Points on that line make the left and right sides exactly equal, so they are not included in the solution region.

How do you graph a system of linear inequalities?

Graphing a system of linear inequalities means finding the region on a coordinate plane where all the inequalities in the system are true at the same time. You can think of it like finding a meeting spot that satisfies everyone's preferences: one person wants to be north of the river, and another wants to be east of the highway. The solution is the overlapping area that makes both people happy.

To do this mathematically, you will graph each inequality one by one on the same set of axes. The final answer isn't a single point or a single line, but rather the entire shaded region where the individual shaded areas overlap.

What is a system of linear inequalities?

A system of linear inequalities consists of two or more linear inequalities working together. Unlike a system of linear equations, which usually has a single intersection point as its solution, a system of inequalities has a whole region of solutions. Any coordinate point $(x, y)$ that lies in the overlapping shaded region will make every inequality in the system true.

How to graph the system

First, graph the boundary line for the first inequality. If the symbol is $\leq$ or $\geq$ , use a solid line. If the symbol is $<$ or $>$ , use a dashed line. Next, pick a test point not on the line (like $(0, 0)$ ) and plug it into the inequality. If the result is true, shade the side of the line containing your test point; if false, shade the opposite side. Repeat this exact process for the second inequality on the same coordinate plane. Finally, darken the area where the two shadings overlap. This overlap is your solution.

Where students slip up

A common mistake is forgetting to flip the inequality sign when dividing or multiplying by a negative number to isolate $y$ . If you forget to flip the sign, you will shade the wrong side of the line! Another frequent slip-up is using a solid line for a strict inequality ( $<$ or $>$ ). Always double-check your symbols before you draw the line.

Worked through

Graph the system: $y \leq 2x + 1$ and $y > -x - 2$ .

First, graph the boundary line for $y \leq 2x + 1$ . Because of the $\leq$ sign, draw a solid line with a y-intercept of $1$ and a slope of $2$ . Test the point $(0, 0)$ : $0 \leq 2(0) + 1$ is true, so shade the region below the line. Next, graph the boundary line for $y > -x - 2$ . The $>$ sign means we draw a dashed line with a y-intercept of $-2$ and a slope of $-1$ . Test $(0, 0)$ : $0 > -(0) - 2$ becomes $0 > -2$ , which is true. Shade the region above the dashed line. Finally, locate the area where the two shaded regions overlap. This overlapping wedge-shaped region is the final solution to the system.

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Where this comes from: OpenStax Intermediate Algebra, Chapter 4: Graphs and Systems · Khan Academy: Systems of inequalities