Graphing Systems of Inequalities
Graph two or more linear inequalities on the same coordinate plane and identify the solution region as the intersection of the half-planes. Apply to real-world constraint problems like budget planning.
A lesson of Algebra 1, in Systems of Equations & Inequalities, about 30 minutes.
Systems of Inequalities
By the end of this lesson, you will be able to graph a linear inequality as a region, find where two regions overlap, test whether a point is a solution, and read what the overlap means in a real problem.
An Inequality Is a Region, Not a Line
$$y = 2x + 1 \quad \text{is a line}$$
$$y > 2x + 1 \quad \text{is everything above it}$$
Every point on the line satisfies the equation. Every point in the shaded half of the plane satisfies the inequality, and there are infinitely many of them in two directions rather than one.
Three steps to graph one:
- Draw the boundary as if it were an equation
- Dashed or solid? Dashed for $<$ and $>$, because the line itself is not included. Solid for $\leq$ and $\geq$, because it is. Same open and closed circle idea as a number line, stretched into two dimensions
- Shade the side that works, and find out which side by testing a point
The Test Point Settles It
Do not reason about which side. Test one.
$$y > 2x + 1$$
Try $(0, 0)$, because the arithmetic is trivial:
$$0 > 2(0) + 1 \longrightarrow 0 > 1 \quad \text{false}$$
So $(0,0)$ is NOT in the region, and the shading goes on the other side.
Use $(0,0)$ every time it is available, which is whenever the boundary does not pass through the origin. If it does, pick any other easy point.
The shortcut people reach for, "$>$ means shade above", works for $y$ on the left and fails the moment the inequality is written any other way. Testing a point never fails.
Example 1: One Inequality
Here is the whole method on one, worked all the way through. $$y \geq -x + 3$$
- Is the boundary line dashed or solid? Type one word solid
- Test the point (0, 0). What does the right side come to? 3
- Is 0 greater than or equal to 3? Type yes or no no
- The test failed, so which side gets shaded? Type "the other side" the other side
Plot It: The Boundary
Two Inequalities: Shade Both, Keep the Overlap
A system of inequalities asks for the points satisfying all of them, so the solution is where the shaded regions overlap.
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