--- title: "Graphical Inequalities and Regions in the Plane" description: "A linear inequality in two variables picks out a half-plane, and several of them together leave a region whose corners answer the question being asked. Drawing the boundary as solid or dashed is what " canonical: https://lightmysky.com/learn/mathematics/graphical-inequalities-and-regions-in-the-plane-mt_sKerzcAcNy source: https://lightmysky.com/learn/mathematics/graphical-inequalities-and-regions-in-the-plane-mt_sKerzcAcNy.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Graphical Inequalities and Regions in the Plane A linear inequality in two variables picks out a half-plane, and several of them together leave a region whose corners answer the question being asked. Drawing the boundary as solid or dashed is what records whether the boundary itself counts. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/graphical-inequalities-and-regions-in-the-plane-mt_sKerzcAcNy ## Ready when they can - Shades the region satisfying y < 2x + 1 and says why the boundary line is dashed - Finds the region satisfying three inequalities at once and lists the integer points inside it - Writes down the inequalities that describe a region already shaded on a grid ## Lesson: Shading the side that passes the test An inequality in two variables gives you a whole half plane, not a line. The boundary line y equals 2x plus 1 splits the grid, and every point on one side passes while the other side fails. Pick a test point such as (0, 0), which is off the line, and shade the side that works. Line, test, shade: that is the whole game. The line style carries meaning. A strict inequality like y less than 2x plus 1 excludes its boundary, so you draw it dashed. An inclusive one like y less than or equal to 2x plus 1 includes it, so you draw it solid. One more trap: multiplying or dividing by a negative flips the symbol, so watch each step as you rearrange. **Example.** Stack three constraints and shade each on the same grid: x at least 0, y at least 0, and x plus y at most 3. Keep only the overlap where every test passes, a small triangle. Hunt integer points by listing pairs: x equals 0 gives 4 points, x equals 1 gives 3, x equals 2 gives 2, x equals 3 gives 1. That is 10 points in total. **Tip.** Reading backwards uses the same skills in reverse. Given a shaded polygon, find each edge line, decide dashed or solid from whether the edge counts, then test a point for the direction. A region on or below the line y equals 4 minus x, edge included, is y less than or equal to 4 minus x. **Recap.** Draw the boundary with the right dash, test a point, and shade or read the side that passes. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB) - [Solving Linear Inequalities and Solution Sets](https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM)