Solving Absolute Value Inequalities: Step-by-Step Guide
Absolute value tells you how far a number is from 0. An absolute value inequality compares that distance to a number. The key rules: |X| < k (with k positive) means -k < X < k. |X| > k means X < -k or X > k. Always get the absolute value by itself first, and check if the other side is negative, which can give a quick answer.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Get the absolute value by itself. Add, subtract, multiply, or divide to get |...| alone on one side.\[|f(x)| \;\square\; k\]
- Check for special cases. If |something| is less than a negative number, there is no solution. If |something| is greater than a negative number, every real number works. Otherwise, keep going.\[|f(x)| < -k \;\Rightarrow\; \varnothing,\quad |f(x)| > -k \;\Rightarrow\; \mathbb{R}\]
- Turn it into a compound inequality. For < or <=, write -k <= expression <= k (AND). For > or >=, write expression <= -k or expression >= k (OR).\[|f(x)|<k \;\Leftrightarrow\; -k<f(x)<k\]
- Solve. Solve like any other inequality. For AND, work on all three parts. For OR, solve each piece separately.\[f(x) = ax + b \Rightarrow x = \frac{f(x) - b}{a}\]
- Write the answer and check. Use interval notation. Test a number inside and a number outside the answer.\[[a, b]\;\text{or}\;(-\infty, a]\cup[b, \infty)\]
Worked example
Solve and write in interval notation: |2x - 1| <= 5
Step 1: The absolute value is already by itself: |2x - 1| <= 5.
Step 2: The right side is positive, so there are solutions.
Step 3: Turn it into an AND inequality: -5 <= 2x - 1 <= 5.
Step 4: Add 1 to all parts: -4 <= 2x <= 6. Then divide all parts by 2: -2 <= x <= 3.
Step 5: In interval notation: [-2, 3].
Check: Try x = 0 (inside): |2(0) - 1| = 1, and 1 <= 5 is true. Try x = 4 (outside): |2(4) - 1| = 7, and 7 <= 5 is false.
Tips and common mistakes
- Always get the absolute value by itself before you split it up.
- Less than becomes AND: |X| < k means -k < X < k. Greater than becomes OR: |X| > k means X < -k or X > k.
- Learn the special cases: |X| < a negative number has no solution, and |X| > a negative number works for every number.
- |X| = 0 has exactly one solution, X = 0.
- If you divide by a negative number inside, remember to flip the inequality sign. Absolute value bars do not stop this.
- Solve an OR as two separate problems and then combine the answers.
- Always test a number to check your answer. It quickly shows if you mixed up AND and OR.