--- title: "Linear Inequalities and Written Region Descriptions" description: "Interpret boundary styles and solution sides from linear inequalities and written region descriptions. Check coordinate pairs against several constraints, count integer-coordinate solutions, and write" canonical: https://lightmysky.com/learn/mathematics/linear-inequalities-and-written-region-descriptions-mt_sKerzcAcNy source: https://lightmysky.com/learn/mathematics/linear-inequalities-and-written-region-descriptions-mt_sKerzcAcNy.md retrieved: 2026-11-12 --- > **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`. # Linear Inequalities and Written Region Descriptions Interpret boundary styles and solution sides from linear inequalities and written region descriptions. Check coordinate pairs against several constraints, count integer-coordinate solutions, and write inequalities for described regions. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/linear-inequalities-and-written-region-descriptions-mt_sKerzcAcNy ## Ready when they can - Use inequality signs, substitution and rearrangement to explain boundary inclusion and which side of a line satisfies a linear inequality - Check whether coordinate pairs satisfy every inequality in a system and count its integer-coordinate solutions - Translate given boundary equations and written descriptions of sides and boundary inclusion into inequalities ## Lesson: Checking points and describing inequality regions You have plotted lines and solved one-variable inequalities. Here you will use equations and written region descriptions. A solution is a pair (x, y). For Ax + By < C, A and B are not both zero; Ax + By = C is the boundary. Check a point by substituting its coordinates: (1, 0) in y < 2x + 1 gives 0 < 3, which is true. A boundary point makes the two sides equal. For < or > it is excluded, so draw the line dashed. For <= or >= it is included, so draw the line solid. For y < 2x + 1, equality on y = 2x + 1 does not pass the strict test. A solid line includes boundary points for that one inequality. **Example.** Rearrange x - 2y >= 8. Subtract x: -2y >= 8 - x. Divide by -2 and reverse the sign: y <= x/2 - 4. Draw the boundary solid and shade below it. The original >= sign does not tell you to shade above. First isolate y or test a point. Choose a test point off the boundary. If it passes, shade its side; if it fails, shade the opposite side. When the origin lies on the line, choose another point. With y isolated, < shades below and > above. The line x = a is vertical: x < a is left and x > a is right. The line y = b is horizontal: y < b is below and y > b is above. Equality changes line style, not which side is shaded. A system requires every inequality to hold. Substitute a point into each one; keep it only if all pass. The overlap may be bounded, unbounded or empty. To count integer-coordinate solutions, fix each allowed integer x and find all allowed integer y-values. For x + y <= k, use y <= k - x. List the pairs column by column, then add the counts. Include a boundary point only when every constraint permits it. *(drawing: For x >= 0, y >= 0 and x + y <= 2, the columns contain 3, 2 and 1 integer points. Total: 6. These bars show counts, not a coordinate graph.)* Translate a written region description using its given boundary equation and allowed side. For y = mx + c, below gives y < mx + c and above gives y > mx + c. Add equality when the boundary is included. For x = a, left gives x < a and right gives x > a. Solid boundaries include equality; dashed ones exclude it. For example, on or below y = 5 - 2x gives y <= 5 - 2x. Apply this to each described boundary. **Example.** For a rectangle with independent x and y limits, count the integer choices for each coordinate and multiply. If -1 <= x <= 1 and 0 <= y <= 1, there are 3 x-values and 2 y-values. Every combination works. Each row below represents one y choice and each column one x choice, giving 2 times 3 = 6 coordinate pairs. **Example.** Strict limits leave their boundary points out. For integer y, y > 0 starts at 1. Consider x >= 0, y > 0 and x + y < 3. Rearrange the last condition to y < 3 - x. At x = 0, y can be 1 or 2. At x = 1, only y = 1 works. At x >= 2, no positive integer y works. Total: 3 points. **Example.** When two sloping limits bound y, compare their lower and upper values in each column. For 0 <= x <= 2, y >= x and y <= 3 - x: at x = 0, y is 0 to 3, giving 4 points; at x = 1, y is 1 to 2, giving 2. At x = 2, no y can be both at least 2 and at most 1. Total: 6 points. **Recap.** Use dashed boundaries for < and >, solid for <= and >=. Test a point off each line. Keep only the overlap, and check every inequality when listing points. *The lesson, practice questions, worked solutions and flashcards were written for this dataset.* ## 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)