Absolute Value Inequalities

An absolute value expression measures the distance of a number from zero on a number line. When we solve absolute value inequalities, we are finding a range of values whose distance from zero meets a certain condition.

To solve these, we remove the absolute value bars by translating the problem into a compound inequality. There are two main types depending on the inequality sign.

1. "Less Than" Inequalities ($<$ or $\le$)

When the absolute value is less than a positive number $c$, the distance from zero is strictly within the range between $-c$ and $c$. This creates an "and" compound inequality.

Rule: If $|A| < c$, then: $-c < A < c$

(Note: The same rule applies to $\le$, just use $\le$ in your compound inequality.)

Example: Solve $|x - 4| < 3$

1. Set up the compound inequality: $-3 < x - 4 < 3$
2. Add 4 to all three parts to isolate $x$: $-3 + 4 < x - 4 + 4 < 3 + 4$ $1 < x < 7$ The solution is all numbers strictly between 1 and 7.

2. "Greater Than" Inequalities ($>$ or $\ge$)

When the absolute value is greater than a positive number $c$, the distance from zero is further away than $c$. This means the expression inside must be either very positive or very negative. This creates an "or" compound inequality.

Rule: If $|A| > c$, then: $A > c \quad \text{or} \quad A < -c$

Example: Solve $|2x + 1| \ge 5$

1. Split into two separate inequalities: $2x + 1 \ge 5 \quad \text{or} \quad 2x + 1 \le -5$
2. Solve the first inequality: $2x \ge 4 \implies x \ge 2$
3. Solve the second inequality: $2x \le -6 \implies x \le -3$ The solution is $x \le -3 \text{ or } x \ge 2$.

A Quick Tip for Memorization

• Less thAND: $<$ or $\le$ translates to an AND inequality (a single, connected range).
• GreatOR: $>$ or $\ge$ translates to an OR inequality (two separate ranges pointing outward).