Solve absolute value equations. | Nonlinear equations in one variable | SAT Math

Absolute value equations split into cases after you isolate them.

核心知识

$| 𝐴 | = 𝐵$ splits into two cases: $𝐴 = 𝐵$ or $𝐴 = - 𝐵$ , but only when $𝐵 \geq 0$ . If $𝐵 < 0$ , there's no solution — absolute value can never be negative.

深入理解

An absolute value equation says distance from zero.

Rule: for $| 𝐴 | = 𝑘$ ,

if $𝑘 < 0$ , no solution
if $𝑘 = 0$ , solve $𝐴 = 0$
if $𝑘 > 0$ , solve $𝐴 = 𝑘$ and $𝐴 = - 𝑘$

When the absolute value is not isolated, isolate it first. For $3 | 2 𝑥 - 1 | + 5 = 20$ , subtract 5 and divide by 3 before you split into cases.

Always check both solutions in the original equation. Most SAT absolute value questions have two valid solutions, but the question might ask for their sum, product, or difference.

分步讲解

Isolate the absolute value expression on one side.
Check whether the other side is negative (no solution), zero (one equation), or positive (two equations).
Write and solve the two cases: expression = positive value, and expression = negative value.
Verify both solutions in the original equation.

常见误解

Forgetting to isolate the absolute value before splitting into cases. $| 𝑥 + 3 | + 2 = 7$ does not split into $𝑥 + 3 + 2 = 7$ and $- (𝑥 + 3) + 2 = 7$ .
Setting up $| 𝐴 | = 𝐵$ as $𝐴 = 𝐵$ and $𝐴 = 𝐵$ (same equation twice) instead of $𝐴 = 𝐵$ and $𝐴 = - 𝐵$ .
Not recognizing that $| 𝐴 | = - 3$ has no solution.

题目

示例解析

How many solutions does the equation $| 3 𝑥 - 6 | + 4 = 2$ have?

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