MathAbsolute Value

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Study Guide▶Math▶Absolute Value
🔢 Algebra~5% of SAT Math

Absolute Value

Master absolute value from fundamentals to advanced applications. Learn to solve equations, inequalities, and interpret absolute value in real-world contexts.

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Worked Example

Practice Problems

Problem

Evaluate: |7|, |-7|, |0|, |-3.5|, and explain why |-x| = |x| for all real numbers.

Problem

Solve each equation and explain your reasoning: a) |x| = 9 b) |x| = 0 c) |x| = -4 d) |x| = 2.5

Problem

Solve completely and verify: |3x - 7| = 11

Problem

Solve: |2x + 3| = |x - 1|

Problem

Solve and graph: |x - 2| > 5

Problem

Solve and graph: |2x + 1| ≤ 7

Problem

Graph y = -2|x + 3| + 5 and identify all key features.

Problem

MANUFACTURING PROBLEM: A machine produces bolts with a target diameter of 8.0 mm. Quality control accepts bolts if they are within 0.3 mm of the target. Write an absolute value inequality and find the acceptable range.

Problem

Write y = |2x - 6| as a piecewise function and graph it.

Problem

SAT PRACTICE PROBLEM: If |2x - 5| = 7, what is the sum of all possible values of x? A) -1 B) 0 C) 5 D) 12

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Watch Out

Common Mistakes

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Thinking |-5| = -5
✔ Remember: absolute value is ALWAYS non-negative
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Confusing |x| with -x
✔ |x| removes negative, -x adds negative
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Forgetting |0| = 0
✔ Zero is the only number whose absolute value equals itself AND its negative
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Forgetting the negative solution
✔ ALWAYS write both x = a AND x = -a
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Thinking |x| = -3 has solutions x = 3 and x = -3
✔ If right side is negative, NO solution exists
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Not checking solutions
✔ Always verify both solutions work in the original equation
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Only solving the positive case
✔ MUST solve BOTH cases: expression = c AND expression = -c
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Forgetting to check solutions
✔ Some equations have extraneous solutions - always verify
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Arithmetic errors when distributing negatives
✔ Be extra careful with signs in the negative case
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Forgetting the A = -B case
✔ MUST consider both A = B AND A = -B
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Squaring without checking for extraneous solutions
✔ Squaring can introduce false solutions - always verify
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Sign errors when distributing the negative
✔ -(x - 1) = -x + 1, not -x - 1
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Writing x > 7 AND x < -3 (impossible!)
✔ Use OR, not AND. A number can't be both > 7 and < -3
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Forgetting to flip inequality when multiplying by negative
✔ Only flip when multiplying/dividing BOTH sides by negative
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Using intersection (∩) instead of union (∪)
✔ Greater than uses UNION (∪) because it's two separate regions
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Splitting into two inequalities like > case
✔ Less than gives ONE compound inequality, not two separate ones
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Forgetting to flip inequality when dividing by negative
✔ If you divide by -2, flip all inequality signs
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Using parentheses instead of brackets for ≤
✔ ≤ includes the endpoint, so use brackets: [-4, 3]
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Thinking y = |x + 3| has vertex at (3, 0)
✔ Vertex is at (-3, 0). The sign INSIDE flips!
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Forgetting that negative a flips the graph upside down
✔ a < 0 means ∩ shape (opens down), not ∪
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Assuming all absolute value graphs pass through origin
✔ Only y = |x| passes through (0, 0). Transformations shift it.
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Writing |x - 8| > 0.3 for tolerance problem
✔ Tolerance uses ≤, not >. You want values WITHIN range.
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Forgetting units in final answer
✔ Always include units (mm, °F, etc.) in real-world problems
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Mixing up target and tolerance
✔ Target is the center (c), tolerance is the allowed deviation (t)
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Forgetting to negate the expression when it's negative
✔ If ax + b < 0, the function is -(ax + b), not just ax + b
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Using wrong inequality signs at critical point
✔ Be consistent: use ≥ on one side, < on the other
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Not simplifying -(ax + b)
✔ Distribute the negative: -(2x - 6) = -2x + 6
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Spending 5+ minutes on one absolute value problem
✔ If stuck after 90 seconds, guess and move on
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Not using answer choices strategically
✔ Plugging in answers is often faster than algebra
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Forgetting to check both solutions
✔ SAT loves to ask for sum/product of ALL solutions
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SAT Strategies

Pro Tips & Shortcuts

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SAT Tip: SAT STRATEGY: When you see |x|, immediately think "this is always ≥ 0." This eliminates wrong answers in multiple-choice questions.

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SAT Tip: SAT TRAP: The SAT loves to include |x| = -k where k > 0. Students rush and solve it, but the answer is "no solution." Always check!

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SAT Tip: SAT PATTERN: If the question asks "How many solutions?", count them. If it asks "What is the sum of all solutions?", add them. If it asks "What is the product?", multiply them.

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SAT Tip: SAT ADVANCED: These problems appear in the last 5 questions of the Math section. If you're short on time, skip and come back.

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SAT Tip: SAT NOTATION: The SAT may ask for the solution in interval notation. Remember to use ∪ (union) to connect the two regions.

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SAT Tip: SAT BRACKETS: Use [ ] for ≤ or ≥ (includes endpoint), use ( ) for < or > (excludes endpoint).

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SAT Tip: SAT GRAPHING: The SAT often shows a graph and asks for the equation. Look for the vertex first, then check if it opens up or down.

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SAT Tip: SAT WORD PROBLEMS: These appear in the last third of the Math section. Read carefully - the SAT tests if you can translate English to math.

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SAT Tip: SAT ADVANCED: These problems appear in the hardest questions. If you see "piecewise" and "absolute value" together, look for the connection.

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SAT Tip: SAT TIMING: Absolute value questions typically appear in positions 10-20 (medium difficulty) and 25-30 (hard). Budget 1-2 minutes each.

⚡ Shortcut

PROFESSOR'S INSIGHT: Think of absolute value as "removing the sign" or "making positive." The key is understanding it measures magnitude, not direction.

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PROFESSOR'S PATTERN: Before solving ANY absolute value equation, check if the right side is positive, zero, or negative. This tells you how many solutions to expect.

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SYSTEMATIC APPROACH: Always write "Case 1:" and "Case 2:" to organize your work. This prevents forgetting the second solution.

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CRITICAL THINKING: When you see |A| = |B|, visualize: "A and B are the same distance from zero." They're either equal or opposites.

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MEMORY TRICK: "Greater than" means "OUTSIDE" - think of the solution regions as being pushed AWAY from the center.

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MEMORY TRICK: "Less than" means "BETWEEN" - think of the solution as being squeezed BETWEEN two boundaries.

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PROFESSOR'S GRAPHING STRATEGY: (1) Plot vertex, (2) Determine direction, (3) Plot one point on each side, (4) Connect with V-shape.

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TRANSLATION GUIDE: "within k units of c" → |x - c| ≤ k. The center is c, the radius is k.

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CRITICAL THINKING: The piecewise form shows WHY the graph has a corner - the function changes from one linear piece to another.

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MASTER STRATEGY: For |ax + b| = c, the sum of solutions is ALWAYS -2b/a. This is a 10-second shortcut for many SAT problems.

⏲ Time Saver

QUICK CHECK: If the answer choices include negative numbers for an absolute value expression, eliminate them immediately.

⏲ Time Saver

TIME SAVER: If you see |something| = negative number, write "no solution" immediately and move on.

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SHORTCUT: If the equation is |ax + b| = c, the solutions will be symmetric around x = -b/a (the value that makes the expression zero).

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PATTERN RECOGNITION: If you see |expression1| = |expression2|, immediately think "two cases: equal or opposite."

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QUICK TEST: Pick a number in your solution region and verify it works. Pick a number outside and verify it doesn't.

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SHORTCUT: For |ax + b| < c, immediately write -c < ax + b < c and solve as one compound inequality.

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QUICK SKETCH: For SAT multiple choice, you often only need to identify vertex and direction - don't waste time finding every detail.

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PATTERN: If problem says "at most k away from c," write |x - c| ≤ k. If it says "at least k away," write |x - c| ≥ k.

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SHORTCUT: For y = |ax + b|, the vertex is always at x = -b/a (where the expression equals zero).

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FINAL TIP: Practice these patterns until they're automatic. On test day, pattern recognition beats calculation speed.

🔢 Calculator

Calculator tip: Most calculators have an abs() function. On graphing calculators, it's usually in the MATH menu under NUM.

🔢 Calculator

Graphing calculator: Use Y= menu and enter abs(x+3) or |x+3| depending on your calculator model.

🔢 Calculator

Calculator strategy: For graphing calculator, graph y = |expression| and y = constant, then find intersection points.

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