Absolute Value Inequalities

Absolute value measures distance from zero. An absolute value inequality bounds how far a quantity can be from zero. Two cases — ≤ and ≥ — produce structurally different solutions.

Definition (piecewise)

For any real number $x$:

The bars strip the sign — they never produce a negative value. This piecewise form is the reason every absolute-value equation or inequality splits into two cases: one where the inside is non-negative (and the bars do nothing) and one where the inside is negative (and the bars flip the sign).

The two cases

Case 1: |expression| ≤ k

This is an intersection (AND). The expression must be within distance $k$ of zero, in either direction. Rewrite as a compound inequality:

Solve all three sections together to isolate the variable.

Case 2: |expression| ≥ k

This is a union (OR). The expression is at least $k$ away from zero, in some direction. Rewrite as two separate inequalities joined by OR:

The solution is two disjoint rays.

Worked example

Solve and graph: $|2x-5|\le 7$.

Step 1 — recognize the structure. This is ≤ (intersection). Rewrite as a compound inequality:

Step 2 — add 5 to all three sections to isolate the variable term:

Step 3 — divide all three sections by 2:

In interval notation: $[-1,6]$.

Common mistakes

1. Dropping the absolute value bars and only solving the positive case. This gives you $x\le 6$ — you’ve lost the lower bound entirely.
2. Forgetting to flip the inequality when multiplying or dividing by a negative number. This rule is non-negotiable; if you divide by $-2$, the inequality direction reverses.
3. Confusing ≤ and ≥ structures. They produce different solution shapes — one bounded interval vs. two unbounded rays.

Machine learning relevance

Absolute value mechanics are the direct mathematical foundation for two cornerstone ML techniques:

• L1 regularization (Lasso): Adds $\sum |{\theta }_j|$ as a penalty term in the cost function, encouraging weights to shrink to exactly zero. This produces sparse models — most features end up with zero weight, leaving you with a small, interpretable subset.
• Mean Absolute Error (MAE): $\frac{1}{m}\sum |y_i-{\hat{y}}_i|$. Unlike MSE, MAE doesn’t square errors, so it’s more robust to outliers. The “absolute value” intuition you build here shapes how you reason about loss functions later.

Understanding the bounds of absolute values is also required for defining margin boundaries in Support Vector Machines and confidence intervals in statistical inference.