Nested absolute value equation solver
||2x - 1| + 3| = 5 has two roots, -1/2 and 3/2: the solver peels the outer absolute value first, keeps only the inner value 2 because -8 is impossible, then solves the inner equation. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 2
- B
- -1
- C
- 3
- D
- 5
- Equation
- ||2x-1|+3|=5
- Inner targets
- 2
- Solution count
- 2
- Solutions
- -0.5, 1.5
Nested absolute value equations put one modulus inside another, so the usual two-case split is not enough: you must resolve the outer absolute value first, keep only non-negative candidates for the inner expression, and only then solve the simple absolute value equation for x. Students meet forms such as the absolute value of the absolute value of ax plus b plus c equals d in intermediate algebra, contest prep, and piecewise-linear models. This nested absolute value equation solver accepts coefficients a, b, c, and d, applies that inside-out case tree deterministically, and returns every valid real root ordered ascending together with the intermediate inner targets. Negative outer right-hand sides and a zero coefficient of x are rejected as invalid input rather than silent empty successes.
Why nested absolute values need an inside-out case tree
A single absolute value equation |ax + b| = k splits into two linear equations when k is positive, one equation when k is zero, and no real solution when k is negative. Nesting a second modulus changes the geometry: the outer absolute value constrains a non-negative quantity u equals the inner modulus, not x itself.
Writing the nested form as ||ax + b| + c| = d therefore begins by setting u = |ax + b| with the hard constraint u ≥ 0, then solving the outer equation |u + c| = d for u. That outer step yields the two candidates u equals d minus c and u equals negative d minus c; any candidate that fails u ≥ 0 is discarded immediately because an absolute value cannot be negative.
Surviving targets become the right-hand sides of ordinary absolute value equations |ax + b| = u. For each valid u the solver produces one root when u is zero and two roots when u is positive, then merges, deduplicates, and sorts all roots ascending. When c is negative the same algebra encodes textbooks that write ||ax + b| − k| = d, because addition of a negative constant is subtraction of k.
When d is zero the two outer candidates collapse to a single condition u equals negative c, which is admissible only if that value is non-negative. This inside-out discipline is exactly what teachers mark on paper and what production APIs need for reproducible grading and notebook scripts.
What you can do with it
Classroom multi-case checks
Let students submit a, b, c, and d for ||ax+b|+c|=d and verify every ordered root against the inside-out case tree taught in intermediate algebra.
Exam and LMS auto-grading
Mark nested absolute value items with a deterministic oracle that rejects negative d, reports empty solutions when no non-negative inner target survives, and never invents extra roots.
Notebooks and piecewise constraints
Solve nested modulus constraints in calibration or tolerance scripts without importing a full computer-algebra library for a four-coefficient problem.
FAQ
How do I solve ||ax + b| + c| = d for x?
Send finite numbers a (non-zero), b, c, and d (non-negative).
What happens if d is negative?
An outer absolute value cannot equal a negative number, so the equation has no real solution and is not billed.
Why must a be non-zero?
If a is zero, ||ax+b|+c|=d is not an equation in x—only a true or false statement about ||b|+c| and d. The solver requires a≠0 so it can return roots for x.
Can a nested absolute value equation have no solution at all?
Yes. When both values allowed by the outer absolute value are negative, the inner absolute value can never reach them, so the equation has no real roots and the list of solutions comes back empty.
How many roots can a nested absolute value equation have?
Each inner value that survives gives one root if it is zero and two if it is positive. With two surviving values, all roots are merged, duplicates are removed and the result is sorted from smallest to largest.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/solve-absolute-value-equation-nested \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":2,"b":-1,"c":3,"d":5}'const res = await fetch("https://api.kit.forhosting.com/algebra/solve-absolute-value-equation-nested", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 2,
"b": -1,
"c": 3,
"d": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/solve-absolute-value-equation-nested",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 2,
"b": -1,
"c": 3,
"d": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/solve-absolute-value-equation-nested", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":2,"b":-1,"c":3,"d":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":2,"b":-1,"c":3,"d":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/solve-absolute-value-equation-nested", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 2,
"b": -1,
"c": 3,
"d": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.solve_absolute_value_equation_nested",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
The same pure module powers the free browser path and the prepaid API, so classroom checks and automation never disagree on which roots survive both layers of absolute value.
Why nested absolute values need an inside-out case tree
When d is negative the outer distance would have to be negative, which is impossible, so the request fails with invalid_input instead of inventing roots.
Input contract, response fields, and error behaviour
Example: a equals two, b equals negative one, c equals three, d equals five yields ||2x-1|+3|=5. The outer peel keeps u equals two (from five minus three) and discards the negative candidate negative eight, then |2x-1|=2 produces the roots x equals negative one half and x equals three halves. POST a JSON body with four required finite numbers: a, the non-zero coefficient of x inside the inner modulus; b, the constant term inside that modulus; c, the constant added to the inner absolute value before the outer modulus; and d, the non-negative outer right-hand side. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, non-finite values, a equal to zero, and d less than zero fail with invalid_input naming the problem, and those failures are not billed. On success the body echoes a, b, c, and d; an equation string that rewrites the nested modulus form for display; inner_targets as the ordered non-negative u values kept after the outer peel; solution_count; solutions as an ascending array of real roots rounded to twelve decimal places for stable goldens across runtimes; and a formula string restating the two-layer identity. If every outer candidate fails the non-negativity filter, the equation is well-formed but has no real root: solution_count is zero and solutions is an empty array rather than an error, so auto-graders can distinguish impossible geometry from bad parameters. Pair the response with your own graphing if you need the V-shaped compositions; this endpoint solves for x only.
Where a nested absolute value API earns its keep
Classrooms and tutoring platforms use a nested absolute value equation solver so learners can check multi-case homework without dropping a sign when peeling the outer modulus first. Exam-prep generators and LMS plugins need a deterministic oracle that rejects negative d as no solution and returns every surviving root when both layers admit solutions, never inventing extras from discarded negative targets. Failed validation does not consume balance. Engineering notebooks reduce piecewise-linear tolerance bands and folded distance constraints to nested modulus equations when calibrating sensors or validating symmetric error models; calling a tiny parse API avoids pulling a full computer-algebra system for four coefficients. Keep requests small, pin the rounded solutions and inner_targets in tests, and you get reproducible roots for homework, contest bundles, and edge workers without standing up symbolic math infrastructure of your own.
Technical questions
Can the solver return zero solutions without an error?
Yes. When d≥0 and a≠0 but every outer candidate for u is negative, solution_count is 0 and solutions is []. That is a well-formed equation with no real root, not invalid input.
What does the nested absolute value equation solver cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.