Absolute value questions trip up a lot of students because they require you to think about two cases at once. Once you learn the pattern, these become some of the easiest points on the test.
What Is Absolute Value?
The absolute value of a number is its distance from zero on the number line. It's always non-negative.
Key insight: means could be or .
Solving Absolute Value Equations
Step 1: Isolate the absolute value expression.
Step 2: Set up two cases — one positive, one negative.
Step 3: Solve each case.
Step 4: Check both solutions in the original equation.
Example 1: Solve
Case 1:
Case 2:
Both solutions check out. Answer: or
Example 2: Solve
No solution! Absolute value can never equal a negative number. The SAT tests this concept regularly.
Absolute Value Inequalities
Here's the rule that makes everything click:
- means (the "AND" case — between)
- means or (the "OR" case — outside)
Memory trick: Less thAND, greatOR
Example 3: Solve
Example 4: Solve
Case 1:
Case 2:
SAT-Style Application
Example 5: A machine fills bottles with 16 oz of juice. The quality control accepts bottles where the actual amount satisfies . What is the acceptable range?
Bottles between 15.5 oz and 16.5 oz pass inspection.
Practice Problems
Problem 1: Solve
Solution
Case 1:
Case 2:
Problem 2: Solve
Solution
Problem 3: How many solutions does have?
Solution
Zero solutions. Absolute value is never negative.
Key Takeaways
- Absolute value = distance from zero, always
- For equations: split into two cases (positive and negative)
- For inequalities: Less thAND ( → between), greatOR ( → outside)
- If the absolute value equals a negative number, there's no solution
- Always check your answers — extraneous solutions can sneak in
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