SAT-85: Mastering Desmos III — Inequalities and Bounded Regions

SAT-85: Mastering Desmos III — Inequalities and Bounded Regions

Description: Desmos shades inequalities automatically. For a system of inequalities, the answer is the region where all the shadings overlap. This lesson shows how to read that region and test whether a point is a solution.

How Desmos shows inequalities

Type an inequality (using <, >, ≤, ≥) and Desmos shades every point that makes it true. With several inequalities, the overlap of all the shaded zones is the solution region — sometimes a closed "bounded region." (Oʻzbekcha: bir nechta tengsizlik boʻlsa, hamma soyalar kesishgan joy — yechim sohasi.)

Reading the boundary lines

• ≤ or ≥ → the boundary line is solid (points on the line count).
• < or > → the boundary line is dashed (points on the line do not count).

(Oʻzbekcha: ≤ va ≥ — chegara chizigʻi qattiq; < va > — uzuq-uzuq.)

The method

1. Type each inequality on its own line.
2. Find where all shadings overlap — that's the solution set.
3. To test a point, plug it into each inequality, or see if it sits in the overlap.

Worked Example 1 — test a point

Is (1, 1) a solution to the system y > x and y < 3?

• y > x: 1 > 1 is false. The point fails the first inequality.
• So (1, 1) is not in the overlap → not a solution.

Worked Example 2 — find a point in the region

Give a point that satisfies y ≥ 0, y ≤ x, and x ≤ 4.

• Try (3, 1): y ≥ 0 ✓ (1 ≥ 0); y ≤ x ✓ (1 ≤ 3); x ≤ 4 ✓ (3 ≤ 4).
• So (3, 1) is in the bounded region.

(Oʻzbekcha: nuqtani har bir tengsizlikka qoʻyib, hammasiga mos kelishini tekshiramiz.)

Worked Example 3 — solid vs dashed

For y ≤ 2x + 1, is the point (0, 1) on the boundary a solution?

• (0, 1): 1 ≤ 2(0) + 1 → 1 ≤ 1 is true. Because the sign is ≤, the boundary line is solid.
• So (0, 1) is a solution (it lies on the included line).

Tip: the "maximum/minimum" of something in a bounded region usually occurs at a corner of the region. Check the corners. (Oʻzbekcha: chegaralangan sohada eng katta/kichik qiymat odatda burchaklarda boʻladi.)

Worked Example 4 — a real-world constraint problem

A student has at most $12 to spend; pens cost $2 (x of them) and notebooks cost $3 (y of them), and they can't buy negative amounts. Write the system and name one valid purchase.

• Constraints: 2x + 3y ≤ 12, x ≥ 0, y ≥ 0. Graph these and the overlap is the bounded region of affordable combinations.
• One valid point: (3, 2) → cost 2(3) + 3(2) = 12 ≤ 12 ✓. So 3 pens and 2 notebooks works.

(Oʻzbekcha: real masalalarda tengsizliklar "cheklov" boʻladi; soyalar kesishgan joy — mumkin variantlar.)

Why the answer is a whole region

Unlike an equation, which usually has a few exact solutions, a system of inequalities is satisfied by infinitely many points — the entire shaded overlap. That's why SAT questions ask things like "which point is a solution?" or "what is the maximum of x + y in the region?" rather than "solve it." Keep that in mind: you're looking for points inside a zone, not a single coordinate. (Oʻzbekcha: tengsizliklar sistemasi cheksiz koʻp nuqtaga ega — butun soyalangan soha yechimdir.)

Practice 1

Is (2, 5) a solution to y > 2x and y < 10?

Practice 2

Does the point (4, 4) satisfy y ≤ x and y ≥ 1?

Key words — Kalit soʻzlar

• Inequality — tengsizlik
• System of inequalities — tengsizliklar sistemasi
• Shaded region — soyalangan soha
• Overlap — kesishish (ustma-ust)
• Bounded region — chegaralangan soha
• Boundary line — chegara chizigʻi
• Solid / Dashed — qattiq / uzuq-uzuq
• Corner (vertex) — burchak
• Satisfy — qanoatlantirish

Summary

• The solution of a system of inequalities is the overlap of all shaded regions.
• ≤/≥ → solid (included) boundary; </> → dashed (excluded).
• Test a point by checking it in every inequality.
• Max/min over a region usually sits at a corner.