Absolute Value Equations & Inequalities Solver
Enter any absolute value equation or inequality and get the full step-by-step solution — two-case split, inequality direction, and a check for extraneous solutions.
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Solving an absolute value equation — two cases worked step by step
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Interpret the absolute value
|2x − 3| = 7 means the expression 2x − 3 is exactly 7 units from zero. This gives two cases.
- 2
Case 1 — expression equals +7
- 3
Case 2 — expression equals −7
- 4
Check both solutions in the original equation
x = 5: |2(5) − 3| = |7| = 7 ✓ | x = −2: |2(−2) − 3| = |−7| = 7 ✓
Final Answer
Absolute value equations and inequalities — the two-case method explained
The absolute value |x| measures the distance of x from zero on the number line — it is always non-negative. This distance interpretation is the key to solving both absolute value equations and inequalities.
To solve |f(x)| = k with k > 0, split into two cases: f(x) = k and f(x) = −k. Solve each independently, then substitute both back into the original to discard extraneous solutions. If k < 0 the equation has no solution; if k = 0 the only solution is f(x) = 0.
Absolute value inequalities introduce direction: |f(x)| < k means −k < f(x) < k (a compound AND inequality, solved as a number-line interval); |f(x)| > k means f(x) < −k or f(x) > k (an OR inequality with two separate regions). The common mistake is forgetting to reverse the inequality sign in the negative case. Absolute value problems appear in SAT Algebra, Common Core, GCSE Higher and Class 11 CBSE.
Absolute value problems to practise (SAT, GCSE, Class 11)
Tap any problem to solve it with full step-by-step working.
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1. |3x + 1| = 10
Absolute value equation — two casesClass 10 / GCSE / SATEasy - Solve with AI →
2. |x - 4| < 6
Absolute value inequality — AND (interval)Class 11 / SATEasy - Solve with AI →3.Absolute value inequality — OR (two regions)Class 11 / SATMedium
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4. |x^2 - 5| = 4
Absolute value of a quadraticClass 11 / A-LevelMedium - Solve with AI →5.Word problem — toleranceSAT / Applied MathsMedium
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Algebra Quiz
Timed MCQs on absolute value, inequalities and linear equations.
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Algebra Flashcards
Drill the two-case rule and inequality direction with spaced-repetition cards.
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Study Planner
Plan your algebra revision — equations, inequalities and absolute value.
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Frequently asked questions
What does absolute value mean and how do I interpret it geometrically?+
The absolute value |x| is the distance from x to zero on the number line — always a non-negative number. So |−5| = 5 and |5| = 5 because both are 5 units from zero. This distance interpretation is the clearest way to understand why |f(x)| = k splits into two cases: f(x) is either 7 units to the right of zero or 7 units to the left.
How do I solve an absolute value equation step by step?+
Write down the two cases: (1) f(x) = k and (2) f(x) = −k. Solve each linear equation separately. Finally, check both answers by substituting back into the original absolute value equation — this catches any extraneous solutions that crept in if f(x) is non-linear.
How do absolute value inequalities work — why are there two different forms?+
When the absolute value is less than k (|f(x)| < k), you need f(x) to stay within k units of zero, giving the compound inequality −k < f(x) < k — a single connected interval. When the absolute value is greater than k (|f(x)| > k), you need f(x) to be more than k units from zero, giving f(x) < −k or f(x) > k — two separate regions. The direction of the inequality determines which form to use.
What are extraneous solutions in absolute value equations and how do I spot them?+
Extraneous solutions appear when the expression inside the absolute value is non-linear (a quadratic, for example). After solving both cases, substitute each answer back: if the original equation gives a true statement, keep it; if not, discard it. For linear expressions like |2x − 3| = 7, extraneous solutions never arise because both cases always produce valid answers.
What is the difference between |x| < k and |x| > k on a number line?+
|x| < k gives the interval (−k, k) — all points within k units of zero, including zero itself. |x| > k gives two rays (−∞, −k) ∪ (k, +∞) — all points more than k units from zero. A quick sketch of a number line with two marks at −k and +k makes both cases immediately obvious.
Are absolute value equations and inequalities on the SAT, GCSE and CBSE syllabus?+
Yes. SAT Math (College Board) includes absolute value equations and inequalities in the Algebra domain. GCSE Higher covers absolute value in the context of number and inequalities. CBSE Class 11 introduces absolute value inequalities in the linear inequalities chapter, and it appears again in Class 12 continuity problems.
Is there a free absolute value calculator that shows the working?+
Yes — this solver shows every step: the two-case split, solving each linear equation, and checking for extraneous solutions. The free plan gives 5 daily solves with no credit card needed. Inequalities, equations, and absolute value of quadratics are all covered.
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