方程与不等式

考什么 · What it tests

需要先会 · Prerequisites

• 移项与合并同类项 / Moving terms and combining like terms
• 用代数式表示未知量 / Expressing unknowns with algebra
• 比与比例 / Ratios and proportions

常见套路 · Common moves

1. 设未知数就设「问什么」：一般把所求量设成 $x$，再把其他量用 $x$ 表示，按题里的相等关系列一条方程——列方程题的第一反应。

   Let the unknown be what's asked: usually set the target quantity as $x$, express the rest in terms of $x$, and write one equation from the stated equality — the first instinct for setting up equations.
2. 不定方程用整数约束收网：一个方程两个未知数时，靠「必须是正整数」「有范围」逐个试，解是有限的。

   Corner a Diophantine equation with integer constraints: with one equation and two unknowns, use "must be a positive integer" and range limits to test values — the solutions are finite.
3. 不等式两边同乘（除）负数要变号：这一步一变号，解集方向整个翻过来。

   Multiplying or dividing an inequality by a negative flips the sign: that single flip reverses the direction of the whole solution set.

易错点 · Common pitfalls

• 不等式两边乘（除）负数忘了变号：-2x < 6 解出 x > -3，方向要翻。

  Forgetting to flip when dividing an inequality by a negative: -2x < 6 gives x > -3 — the direction reverses.
• 设了未知数却漏用某个条件：题目给的每个数通常都要进方程，少用一个条件基本无解或多解。

  Setting up the unknown but leaving a condition unused: nearly every given number belongs in the equation, and skipping one leaves it unsolvable or ambiguous.