Absolute Value Inequalities

An absolute value inequality — , — is solved by turning it into a compound inequality and then using the method from Compound Inequalities. Which compound you get depends entirely on the direction of the inequality symbol.

• (less than) → the inside is close to zero → the “and” compound → one interval.
• (greater than) → the inside is far from zero → the “or” compound or → two pieces.

As with Absolute Value Equations, the first move is always to isolate the absolute value.

The Absolute Value Inequality Formula

Both translations require . The versions with and are identical with the endpoints included.

Why the Translation Works

Read as “the distance from to .” Then asks: which numbers sit less than units from zero? Those are exactly the numbers between and , which is the statement . The condition asks the opposite: which numbers sit more than units from zero? Those are the numbers to the left of together with the numbers to the right of , which is or .

You can also get there from the piecewise definition. If then , so becomes ; if then , so becomes , i.e. . Combining the two cases gives . The same case split on produces the two-piece “or” result. Either way, the compound inequality is not a rule to memorize blindly — it is the distance idea written in symbols.

Reading the Answer on a Number Line

The graph makes the “and” versus “or” distinction visible. For you shade the single segment between the two boundary points and use open dots (or closed dots for ); the picture is one unbroken piece straddling the center. For you shade two rays heading in opposite directions, away from the center, leaving a gap in the middle. If your algebra produces one interval for a “greater than” problem, or two pieces for a “less than” problem, the graph will look wrong at a glance — a quick sanity check before you commit to an answer.

The “Less Than” Case: An Interval Around Zero

Interval: . One connected piece, centered where the inside is zero.

The “Greater Than” Case: Two Pieces Away From Zero

Interval: . Two pieces, everything except the middle.

Special Cases

Worked Example A: “Less Than”

Solve and write the answer in interval notation.

Step 1 — isolated already. Translate to an “and” compound:

Step 2 — subtract from all three parts:

Step 3 — divide all three parts by :

Answer: .

Worked Example B: “Greater Than”

Solve .

Translate to an “or” compound:

Solve each:

Answer: .

Worked Example C: Isolate First

Solve .

Step 1 — isolate the absolute value:

Step 2 — translate to an “or” compound (, so include endpoints):

Step 3 — solve each:

Answer: .

Worked Example D: The Special Cases

Solve . An absolute value is never negative, so it can never be less than . Answer: .

Solve . An absolute value is always , which is always greater than . Answer: .

Solve . Nothing has a negative absolute value, and excludes zero itself. Answer: .

Worked Example E: A Tolerance Problem

A machine shop cuts rods to a target length of mm and accepts any rod whose length is within mm of the target. Write the acceptance condition as an absolute value inequality, then as an interval of allowed lengths.

Step 1 — “within of ” is a distance statement:

Step 2 — translate the “less than or equal” form to an “and” compound:

Step 3 — add to all three parts:

Answer: a rod is accepted when mm. A rod failing the spec by more than the tolerance satisfies , i.e. or .

Common Mistakes to Avoid

• Using the wrong compound. “Less than” → “and” (one interval); “greater than” → “or” (two pieces). Swapping them gives an answer that is the complement of the correct one.
• Writing a “greater than” answer as a single interval. is , never .
• Not isolating first. must become before translating.
• Forgetting to negate the whole side in the “or” case. gives , not with a sign dropped.
• Missing the always/never special cases. An absolute value compared to a negative is either always true or never true — no algebra needed.
• Bracket errors. / include endpoints (brackets); / exclude them (parentheses).

Where This Shows Up Later

• Tolerances and error bounds. “Within mm of mm” is ; “off by more than ” is .
• Distance conditions. describes the points within distance of — a neighborhood, used constantly in later math.
• Limits (the – definition). whenever is two absolute value inequalities.
• Domains and constraints. Some domain restrictions and feasibility conditions are absolute value inequalities.

Practice Problems

Work each problem yourself before opening the answer. Give solutions in interval notation.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. A thermostat holds a room within of a setpoint. Write the allowed temperature range as an interval.

Quick Reference

The compound you produce is solved with Compound Inequalities; the answer format is Interval Notation. The equation version is Absolute Value Equations. More Algebra lessons are available too.