Absolute Value Equations & Inequalities
SAT Math — Complete Mastery Guide
Algebra
SAT Core Topic
High-Frequency
1 What Is Absolute Value?
1.1 Core Idea — Distance from Zero
The absolute value of a number $x$, written $|x|$, is the distance between $x$ and $0$ on the number line. Because distance is never negative, absolute value is always non-negative:
$$|x| \ge 0 \quad \text{for every real number } x$$
Notation: We use vertical bars around an expression: $|x|$, $|2x-3|$, $|a-b|$.
1.2 Geometric Meaning
- $|x|$ = distance from $x$ to $0$.
- $|a - b|$ = distance between $a$ and $b$ on the number line.
1.3 Why Absolute Value Matters
Absolute value measures magnitude regardless of sign. It appears in:
- Distance — how far apart two values are
- Error / Tolerance — how far a measurement is from the target
- Deviation — how far a data point is from the mean
Interactive visualization — available in the full lesson
Drag the red point along the number line. The blue dashed segment shows the distance from 0 — that's $|x|$.
2 Formal Definition
$$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$
In words: If $x$ is zero or positive, absolute value leaves it alone. If $x$ is negative, absolute value flips the sign (multiplies by $-1$) to make it positive.
$|5| = 5$
5 is positive → keep it
$|-7| = 7$
$-7$ is negative → $-(-7)=7$
$|0| = 0$
0 is neither → $|0|=0$
Key Insight: The "$-x$" in the definition does not mean "negative." It means "the opposite of $x$." If $x = -3$, then $-x = -(-3) = 3$, which is positive.