Linear Inequalities & Graphs — 700-800

The 700-800 Patterns: Linear Inequalities & Graphs

Part 1 of 3 — Constraint Systems Under Pressure

Hard-tier inequality items are optimization problems in disguise: multiple constraints, and the answer lives at whichever constraint binds first.

Archetype 1: The Binding Constraint

When two resources limit production (beads AND clasps, rods AND sheets), compute the limit each imposes separately — the smaller limit wins. The larger limit is always an option.

Worked example. Each birdhouse needs $5$ boards and $8$ screws; $120$ boards and $176$ screws available. Max birdhouses?

Boards: $120 \div 5 = 24$. Screws: $176 \div 8 = 22$. The screws bind: $22$. An item may then ask for revenue at \20$each:$22 \times 20 = $440$— and plant$24 \times 20 = $480$plus the raw count$22$ as options.

Archetype 2: Substitute the Known Coordinate, Combine the Bounds

Given a system like $y < -\frac{2}{3}x + 8$ and $y \ge x - 4$ with a point $(6, b)$: substitute $x = 6$ into BOTH inequalities to get a bound from each, then take the tighter one. Strict inequalities exclude their boundary — "greatest integer $b$" with $b < 4$ is $3$, not $4$.

Worked example. $(k, 9)$ satisfies $y \le 4x + 3$ and $y \ge -2x + 15$. Least integer $k$?

First: $9 \le 4k + 3$, so $k \ge 1.5$. Second: $9 \ge -2k + 15$, so $k \ge 3$. Both must hold — the stricter bound $k \ge 3$ governs: $k = 3$. Using only one inequality gives $2$, the planted trap.

Archetype 3: Requirements Plus a Budget (Round the Right Way)

Real-world systems mix a minimum requirement ("at least 40 chairs", "at least $170 earned") with a cap (budget, total hours). Fund the requirement first, then see what's left — and round DOWN for capacity, UP for requirements.

Worked example. Chairs \12$each (at least$40$required), tables$$45$each, budget$$1{,}500$. Max tables?

Chairs claim 40 \times 12 = \480$, leaving$$1{,}020$. Tables:$1020 \div 45 = 22.67 \rightarrow$**$22$**. Ignoring the chair requirement gives$33$; rounding up gives$23$ — both are in the options.

Traps & Speed: Linear Inequalities & Graphs

Part 2 of 3 — Boundary Games and Test-Point Discipline

The Boundary Is Where They Get You

Nearly every hard inequality distractor is a boundary story:

• Strict boundary planted as an answer. If $3x + 2y < 18$, a point giving exactly $18$ FAILS — but it will be an option, and it will pass the other inequality convincingly.
• Round-up on a budget / round-down on a requirement. Both wrong directions always appear. Budget: $22.67 \rightarrow 22$. Requirement: $8.64 \rightarrow 9$.
• The ignored constraint. Answers computed from only one of the two (or three) conditions are systematically planted — usually as the LARGER, more satisfying number.
• Least vs. greatest. After combining bounds like $2 \le b < 4$, the test asks for one end and offers both.

Test-Point Discipline (the 20-second method)

For "which ordered pair is a solution" items, don't graph — test each option in the harder inequality first (the one more options will fail). Eliminate, then run survivors through the second inequality. Check strict vs. non-strict on EVERY substitution that lands exactly on a boundary; the test engineers one option to land there on purpose.

Desmos Exploits

• Type both inequalities directly (Desmos shades them); the solution region is the overlap. Then check options by clicking — or plot each option as a point and see which lands in the dark region. Points on a dashed boundary don't count; on a solid boundary they do.
• For "greatest integer $b$ such that $(6, b)$ works": add the vertical line $x = 6$ and read off where it crosses the shaded overlap.
• Word problems: translating to a system and letting Desmos shade is usually SLOWER than funding the requirement first and dividing the leftover. Save Desmos for genuinely two-dimensional questions.

Timed Drill: Linear Inequalities & Graphs

Part 3 of 3 — Four Questions, Full Difficulty

Target pace: 75 seconds per question.

Pre-flight checklist:

1. Count the constraints. Every one of them must be used or consciously ruled out.
2. Requirement $\rightarrow$ round up. Budget/capacity $\rightarrow$ round down. Say it before you round.
3. Strict inequality $\rightarrow$ the boundary itself is OUT.

Start the clock.