Linear Equations & Inequalities — 700-800

The 700-800 Patterns: Linear Equations & Inequalities

Part 1 of 3 — The Three Archetypes the Hardest Items Reuse

At the 700-800 level, linear equation questions stop testing whether you can solve — they test whether you notice structure and answer the exact quantity asked. Three archetypes cover nearly every hard item.

Archetype 1: Parameter Equations (No Solution / Infinitely Many)

An equation like $a(2x - 1) + 7 = 6x + 3$ with a constant $a$:

• No solution: the $x$-coefficients match, but the constants don't.
• Infinitely many solutions ("true for all $x$"): the $x$-coefficients match AND the constants match.

Worked example. For what value of $a$ does $a(5x + 2) = 20x + 9$ have no solution?

Expand: $5ax + 2a = 20x + 9$. Match coefficients: $5a = 20$, so $a = 4$. Check constants: $2a = 8 \ne 9$. Constants differ, so $a = 4$ gives no solution. If the question said "true for all $x$," you'd need BOTH $5a = 20$ and $2a = 9$ — impossible here, so no such $a$ would exist.

Speed move: the coefficient equation gives the candidate instantly. Only the constants decide between "no solution" and "infinitely many."

Archetype 2: Solve for $x$... Then Answer Something Else

Hard items almost never ask for the variable you solve for. They ask for the total, the difference, or an expression like $\frac{x + 3}{2}$. The option list always contains your intermediate value — that's the trap.

Worked example. A café sold $6$ more lattes than mochas. Lattes are \5$, mochas are$$4$, total revenue$$174$. How many total drinks?

Let $m$ = mochas: $5(m + 6) + 4m = 174$, so $9m + 30 = 174$, $9m = 144$, $m = 16$. Lattes: $22$. Total: $38$. The options will include $16$ and $22$ — both are bait for someone who stops early.

Archetype 3: Constrained Maximum (Fees, Thresholds, Round DOWN)

"Greatest number of ___ within budget" items add a flat fee, sometimes a conditional fee that only applies past a threshold, then punish rounding errors.

Worked example. Rental: \80$flat plus$$14$per hour; jobs over$10$hours add a$$35$surcharge. Budget$$300$. Max whole hours?

Past 10 hours: $80 + 35 + 14h \le 300$, so $14h \le 185$, $h \le 13.2$, giving $h = 13$. Verify: $115 + 14(13) = 297 \le 300$; $14$ hours would cost $311$. Always round down and verify the boundary.

Traps & Speed: Linear Equations & Inequalities

Part 2 of 3 — How the Wrong Answers Are Built

Every hard-tier option list is engineered. If you know the four molds the distractors come from, you can often eliminate two options before doing any algebra.

The Four Distractor Molds

1. The intermediate value. You solved for $x$ correctly — and $x$ is sitting right there as an option. But the question asked for $2x - 5$, or the total, or the other quantity. Reread the final sentence before you answer.
2. The boundary value. "Costs LESS than" is strict: at the break-even point the totals are EQUAL, which fails "less than." The break-even number itself is always an option.
3. The wrong rounding direction. Budget and capacity problems round down ($22.9 \rightarrow 22$); "at least" requirement problems round up ($8.6 \rightarrow 9$). The opposite rounding is always an option.
4. The sign-flip casualty. Dividing an inequality by a negative reverses it. The un-flipped answer is always an option.

Speed Techniques

• Desmos both-sides graph: for $6x - 8 = 3x + 7$, type $y = 6x - 8$ and $y = 3x + 7$; the intersection's $x$-coordinate is the solution. For parameter problems, add a slider for $a$ and drag until the lines are parallel.
• Boundary check beats algebra: for "greatest $n$" problems, test the two integers around your cutoff. Thirty seconds of arithmetic catches every rounding trap.
• Don't isolate — evaluate. If the target is $\frac{x+3}{2}$, compute it directly from $x$; if the target is a combination like $x + y$, look for a way to build it without finding $x$ and $y$ separately.

Timed Drill: Linear Equations & Inequalities

Part 3 of 3 — Four Questions, Full Difficulty

Target pace: 75 seconds per question. That is the real budget these carry on test day if you want time banked for harder algebra later in the module.

Drill discipline:

1. Read the final sentence first — know the quantity being asked before you touch the setup.
2. After solving, spend 5 seconds asking: "is this the asked-for quantity, or my intermediate?"
3. On max/min questions, verify the boundary integer before committing.

Start the clock.