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A Line Splits the Plane

Part of the Graphs & Systems section of Coddy's Algebra journey. Lesson 42 of 64.

An equation like y = x + 1 has a line as its graph. Change the = to < and the solutions are no longer a line. They fill a whole region. The solutions of y < x + 1 are all the points below the line y = x + 1. At every point below the line, the y-coordinate is smaller than x + 1.

$y < x + 1: everything below the line$

The solutions of y < x + 1 are all the points below the line

The equation gives the boundary line. Once y is isolated on the left, y > mx + b shades above the line and y < mx + b shades below it. The inequality describes all points in that half-plane.

Challenge

Medium

Shade the solutions of y < 2x - 1.

Try it yourself

y < 2x - 1

Draw the graph

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Graphs & Systems

1The Coordinate Plane

Two Numbers, One Point The Four Quadrants Points on the Axes Mirror Points Recap: The Plane

4Slope-Intercept Form

y = mx + b The Starting Height Write the Equation Solve for y Draw From Any Form Recap: y = mx + b

7Parallel & Perpendicular

Same Slope, Never Meet The Perpendicular Slope A Right-Angle Line Recap: Two Relations

10Systems by Graphing

Two Lines, One Answer Solve by Drawing Check the Crossing When Lines Never Cross Recap: By Graphing

2From Rule to Line

A Rule Makes Points The Points Line Up On the Line or Not?Between the Dots Recap: Rule to Line

5Point-Slope Form

A Point and a Slope From Two Points Flat or Vertical Through Draw From a Point Recap: Point-Slope

8Function Notation

The Name of a Rule Inputs With Signs Find the Input The Graph of f Recap: f(x)

11Solving Systems Exactly

Substitution Isolate First Elimination Scale, Then Add Three Unknowns Two Numbers, Two Facts Recap: Two Methods

3Slope

Steepness as a Number Slope From Two Points Downhill Lines Flat and Straight Up A Fraction of a Slope Recap: Slope

6Standard Form

Ax + By = C The Two Intercepts Draw by Intercepts Between the Forms Recap: Standard Form

9Shaded Regions

A Line Splits the PlaneSolid or Dashed Test a Point Rearrange, Then Shade Recap: Half-Planes Two Shadings Overlap The Corner of a Region

12Final Challenges

Final: Round Trip Final: Where They Meet Final: Draw the Fence Final: The Whole Plane

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