SAT — mathMedium6 min read

Graphing Linear Inequalities in Two Variables

Graph linear inequalities in the xy-plane for the Digital SAT. Learn boundary lines, shading direction, and how to identify solution regions.

Tutor AI Team2026-02-25T01:17:12.160Z

Graphing linear inequalities in two variables extends the concept of graphing lines into regions of the coordinate plane. Instead of a single line of solutions, a linear inequality has an entire half-plane of solutions. The Digital SAT tests your ability to identify which region satisfies an inequality and to match an inequality to its graph.

Core Concepts

From Equation to Inequality

The equation $y = 2x + 3$ is a line. The inequality $y > 2x + 3$ represents all points above that line. The inequality $y < 2x + 3$ represents all points below it.

Boundary Lines

Solid line (———): used for $\leq$ or $\geq$ (boundary is included)
Dashed line (- - -): used for $<$ or $>$ (boundary is not included)

Shading Direction

After drawing the boundary line:

If $y > mx + b$ or $y \geq mx + b$ : shade above the line
If $y < mx + b$ or $y \leq mx + b$ : shade below the line

The Test Point Method

If you're unsure which side to shade:

Pick a test point NOT on the line ( $(0, 0)$ is easiest if it's not on the line).
Substitute into the inequality.
If true → shade that side. If false → shade the other side.

Example: $y > 2x - 1$ . Test $(0, 0)$ : $0 > 2(0) - 1$ → $0 > -1$ ✓. Shade the side containing $(0, 0)$ .

Graphing Step by Step

Graph the boundary line ( $y = mx + b$ ).
Determine solid or dashed.
Shade the correct half-plane.

Standard Form Inequalities

For $Ax + By > C$ , solve for $y$ first to determine the shading direction:

$2x + 3y \leq 12$ → $3y \leq -2x + 12$ → $y \leq -\frac{2}{3}x + 4$

Boundary: solid line $y = -\frac{2}{3}x + 4$ . Shade below.

Identifying Inequalities from Graphs

Given a graph with a boundary line and shading:

Find the equation of the boundary line.
Check if the line is solid (≤ or ≥) or dashed (< or >).
Check if shading is above (> or ≥) or below (< or ≤).

Strategy Tips

Tip 1: Always Solve for y First

Converting to $y > mx + b$ form makes it clear which direction to shade.

Tip 2: Use $(0, 0)$ as Your Test Point

It's the easiest point to substitute. The only exception is when the line passes through the origin.

Tip 3: Dashed vs. Solid Is About the Symbol

Strict inequalities ( $<$ , $>$ ) → dashed. Non-strict ( $\leq$ , $\geq$ ) → solid. This is a common answer-differentiator on the SAT.

Tip 4: Points on the Line

If the inequality uses $\leq$ or $\geq$ , points on the boundary line ARE solutions. If it uses $<$ or $>$ , points on the boundary line are NOT solutions.

Tip 5: On the SAT, Read the Graph Carefully

Check whether the line is solid or dashed, and which side is shaded. These details determine the correct inequality.

Worked Example: Example 1

Problem

Graph $y < x + 4$ .

Boundary: $y = x + 4$ (dashed line, slope 1, y-intercept 4)

Shade: below the line.

Test $(0, 0)$ : $0 < 0 + 4 = 4$ ✓ → $(0, 0)$ is in the shaded region.

Solution

Worked Example: Example 2

Problem

Graph $3x + 2y \geq 6$ .

Solve for $y$ : $y \geq -\frac{3}{2}x + 3$

Boundary: solid line, slope $-\frac{3}{2}$ , y-intercept 3.

Shade: above.

Solution

Worked Example: SAT-Style

Problem

Which inequality matches a graph with a dashed line through $(0, 2)$ and $(4, 0)$ with shading below?

Slope: $\frac{0 - 2}{4 - 0} = -\frac{1}{2}$

Equation: $y = -\frac{1}{2}x + 2$

Dashed + below → $y < -\frac{1}{2}x + 2$

Solution

Worked Example: Example 4

Problem

Is the point $(3, 1)$ a solution to $2x - y \leq 7$ ?

$2(3) - 1 = 5 \leq 7$ ✓. Yes, $(3, 1)$ is a solution.

Solution

Worked Example: Example 5

Problem

Graph $y \geq -3$ .

This is a horizontal line at $y = -3$ (solid). Shade above.

Solution

Practice Problems

Problem 1

Graph $y > -2x + 5$ . Is $(0, 0)$ in the solution region?

Problem 2

Graph $x + y \leq 8$ . Is $(3, 6)$ a solution?

Problem 3

A graph shows a solid line with slope 2 and y-intercept $-1$ , with shading above the line. Write the inequality.

Problem 4

Is $(5, 3)$ a solution to $y < \frac{1}{2}x + 1$ ?

Problem 5

Graph $x \geq 2$ in the coordinate plane.

Problem 6

Which of the following points satisfies $3x + 4y < 12$ : $(0, 4)$ , $(2, 1)$ , or $(4, 0)$ ?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Shading the wrong side. Always use a test point to verify.
Confusing solid and dashed lines. $\leq$ and $\geq$ → solid. $<$ and $>$ → dashed.
Not converting to $y =$ form before determining shade direction. When dividing by a negative $B$ in standard form, the inequality flips.
Thinking points on a dashed line are solutions. They're not — dashed means the boundary is excluded.
Ignoring the inequality direction when converting from standard form. If you divide $-2y \geq 4$ by $-2$ , you get $y \leq -2$ , not $y \geq -2$ .

Frequently Asked Questions

How do I shade on the Desmos calculator?

Type the inequality directly (e.g., $y > 2x + 1$ ) and Desmos shades the correct region automatically.

What if the question asks for the number of integer solutions in a region?

List or count the integer coordinate points that satisfy all the given inequalities.

Can an inequality have no solution?

A single linear inequality in two variables always has infinitely many solutions (an entire half-plane). But a system of inequalities can have no solution.

How are graphing inequalities different from graphing equations?

An equation gives a line (1D). An inequality gives a region (2D) — a half-plane on one side of the line.

Do I need to graph precisely on the SAT?

For multiple choice, you just need to identify the correct graph or inequality. You won't need to draw a precise graph.

Key Takeaways

✓
Boundary line: graph the corresponding equation first.
✓
Solid line for $\leq$ , $\geq$ ; dashed line for $<$ , $>$ .
✓
Shade above for $y > mx + b$ or $y \geq mx + b$ ; shade below for $<$ or $\leq$ .
✓
Test point method: plug in $(0, 0)$ to determine which side to shade.
✓
Converting to y-form makes shading direction clear.
✓
On the SAT, identifying the correct graph or inequality is more important than drawing one.

Tags:

#sat#math#algebra#inequalities#graphing

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Core Concepts

From Equation to Inequality

Boundary Lines

Shading Direction

The Test Point Method

Graphing Step by Step

Standard Form Inequalities

Identifying Inequalities from Graphs

Strategy Tips

Tip 1: Always Solve for y First

Tip 2: Use (0,0)(0, 0)(0,0) as Your Test Point

Tip 3: Dashed vs. Solid Is About the Symbol

Tip 4: Points on the Line

Tip 5: On the SAT, Read the Graph Carefully

Worked Example: Example 1

Worked Example: Example 2

Worked Example: SAT-Style

Worked Example: Example 4

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?

Tip 2: Use $(0, 0)$ as Your Test Point