Absolute Value Inequality
An absolute value inequality is an inequality that contains an absolute value expression, such as or . Solving one produces a compound inequality — either an "and" inequality (for less-than) or an "or" inequality (for greater-than).
An absolute value inequality is an inequality in which the variable appears inside an absolute value. For expressions of the form (where ), the solution is the conjunction . For expressions of the form , the solution is the disjunction or . These rules also apply when and are replaced by and , respectively.
Key Formula
Where:
- = an algebraic expression (often containing a variable)
- = a positive number representing the boundary value
Worked Example
Problem: Solve and express the solution as an inequality and in interval notation.
Step 1: Since the inequality uses ≤ (less than or equal), rewrite it as a compound "and" inequality.
Step 2: Add 4 to all three parts to begin isolating x.
Step 3: Divide all three parts by 2.
Step 4: Write the solution in interval notation.
Answer: The solution is , or in interval notation, .
Visualization
Why It Matters
Absolute value inequalities show up whenever you need to describe a range of acceptable values around a target. In manufacturing, for instance, a machine part might need to be within 0.02 cm of a specified length — that tolerance is naturally expressed as an absolute value inequality. They also appear throughout statistics when measuring how far data points fall from a mean.
Common Mistakes
Mistake: Using an "and" compound inequality when the absolute value is greater than a number (or vice versa).
Correction: Remember: "less thAND" — if , use an "and" inequality. "greatOR" — if , use an "or" inequality. Mixing these up gives incorrect solution sets.
Mistake: Forgetting to isolate the absolute value expression before splitting into cases.
Correction: If the inequality is something like , first add 2 and divide by 3 to get before applying the rules.