Graph Linear Inequalities in Two Variables

From Intervals to Regions

In this lesson:

• Graph a linear inequality as a half-plane; solid or dashed boundary
• Graph systems; identify the feasible region

What You Will Be Able to Do

1. Graph a linear inequality: solid or dashed boundary; shade the correct half-plane
2. Shade the correct side using the test-point method
3. Write the inequality from a shaded half-plane graph
4. Graph a system and identify the feasible region
5. Verify whether a point satisfies a system

From Number Line to Half-Plane

One variable: interval (a ray). Two variables: half-plane (a region).

Boundary Line Divides the Plane in Two

The boundary creates two half-planes:

• Above:
• Below:

: ✓ — in the solution set.

: ? No — not in the solution set.

Solid vs. Dashed: Boundary Inclusion Rules

A boundary point satisfies equality exactly — strict inequality is false there.

Parallel: open vs. closed circle on the number line.

The Test-Point Method: Three Steps

After drawing the boundary line:

1. Choose a test point not on the boundary — unless the line passes through the origin
2. Substitute the test point into the inequality
3. Shade the half-plane that satisfies the inequality

Don't memorize "shade above/below" — test directly.

Graphing an Inequality in Slope-Intercept Form

Step 1: Draw the boundary . Strict () → dashed line.

Step 2: Test : . True ✓

Step 3: Shade the half-plane containing the origin (below the line).

Graphing an Inequality in Standard Form

Step 1: Draw boundary (intercepts: and ). Non-strict → solid.

Step 2: Test : ? False ✗

Step 3: Shade the half-plane not containing the origin.

Worked Example: Line Through the Origin

Graph . Line passes through — cannot use origin as test point.

Test : ? False ✗

Shade the side not containing — above-left.

Use or when the line passes through the origin.

Common Error: Shading the Wrong Half-Plane

Wrong approach: "The inequality is , so rearrange: . Shade below."

This is correct — but only after rearranging correctly, which students often do wrong.

Reliable approach: Test every time. No rearranging needed.

The test-point method never fails.

Systems: Solution Is the Overlap

For a system of inequalities, a point is a solution iff it satisfies all inequalities simultaneously.

Graph each inequality separately → shade each half-plane → the solution to the system is the intersection of all shaded regions.

Graphing a System of Two Inequalities

and

Ineq. 1: Dashed . Origin satisfies → shade above.

Ineq. 2: Solid . Origin satisfies → shade below.

Feasible region: above and at or below .

Three-Inequality System and Feasible Region

System: and and

Feasible region: the triangle with vertices , , .

Bounded vs. Unbounded Feasible Regions

Bounded: enough constraints to enclose a finite region (polygon).

Unbounded: fewer constraints → region extends to infinity.

Two inequalities typically produce an unbounded region.
Three or more may produce a bounded polygon.

Count constraints but check the graph — not all three-inequality systems are bounded.

Check-In: Graph a Single Inequality

Graph .

1. Draw the boundary — solid or dashed?
2. Test
3. Shade the correct half-plane

Then test : is it in the solution region?

Write the Inequality from the Graph

Solid line through and ; region above-left shaded.

Step 1: Slope , so .

Step 2: Solid line → or .

Step 3: Test : ✓ — but is unshaded. Flip: .

Real-World Constraints: Bakery Production Hours

A bakery makes cakes () and pies (). Each cake: 2 hours; each pie: 3 hours. Total available: 18 hours.

Graph the system. Is feasible? Is feasible?

What You Can Do Now

✓ Solid for /; dashed for /

✓ Test ; shade the satisfying side

✓ System solution = overlap of all half-planes

Dashed for strict — boundary excluded from solution

Test first — don't assume "shade above"

Overlap only — not the union of shaded regions

Coming Up: Modeling with Inequalities

Next: HSA.CED.A.3 — represent constraints as systems of inequalities; interpret feasible regions.

Linear programming: find the best point in the feasible region — optimization under constraints.

REI.D.12 built the tool. CED.A.3 uses it to model and optimize.