Absolute value equation two solutions calculator
|2x − 3| = 5 splits into 2x − 3 = 5, which gives 4, and 2x − 3 = −5, which gives −1. Enter your three numbers to see both case equations and their roots in order, ready to copy into your working. 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
- -3
- C
- 5
- Equation
- |2x-3|=5
- Solution count
- 2
- Solutions
- -1, 4
When you solve an absolute value equation of the form modulus of ax plus b equals c, textbooks insist on writing two separate linear equations: the positive case ax plus b equals c, and the negative case ax plus b equals negative c. That split is the entire method, yet it is easy to drop a sign, forget the second case, or invent a root when the right-hand side is negative. This absolute value equation two solutions calculator accepts coefficients a, b, and c, runs both cases side by side, and returns the ordered real solutions together with the rewritten case equations so you can show the work.
Why absolute value equations produce two candidate solutions
The absolute value of a real number is its distance from zero on the number line. Writing |ax + b| equals c therefore asks for every real x such that the linear expression ax plus b sits exactly distance c away from zero. Distance is unsigned, so the expression may land on the positive side at plus c or on the negative side at minus c.
That geometric reading is exactly the two-case split used in every algebra course: either ax plus b equals c (the positive case), or ax plus b equals negative c (the negative case). Solving each linear equation gives x equals (c minus b) divided by a from the positive case, and x equals (negative c minus b) divided by a from the negative case, provided a is not zero.
When c is strictly positive the two cases yield two distinct real roots, which this calculator returns in ascending order together with both case equations so a learner can verify each branch. When c is exactly zero the distance is zero, the two cases collapse into the single equation ax plus b equals zero, and only one root is returned: x equals negative b over a. When c is negative a distance cannot be negative, so there is no real solution at all.
Coefficient a must be non-zero: if a were zero the original statement would not be a linear equation in x, only a true or false claim about the modulus of b compared with c, which is outside the scope of solving for two candidate roots of x.
Where a two-solutions absolute value calculator earns its keep
Classrooms and tutoring platforms use an absolute value equation two solutions calculator so learners can check both case roots for arbitrary a, b, and c without flipping a sign by hand or forgetting the negative branch. Exam-prep generators need a deterministic oracle that marks wrong the case c less than zero instead of silently returning an empty array that a buggy grader might treat as success, and that always surfaces both positive and negative case equations for partial-credit rubrics. Failed validation does not consume balance.
Keep requests small, pin the rounded solutions and case fields in tests, and you get reproducible dual roots for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom two-case checks
Let students submit a, b, and c for |ax+b|=c and verify both ordered roots plus the positive and negative case equations against textbook methods.
Exam and LMS dual-root grading
Mark absolute value items that require both solutions with a deterministic oracle that rejects negative c as no solution and never invents a second root when c is zero.
Notebooks and symmetric constraints
Solve symmetric linear absolute constraints |ax+b|=c in calibration scripts, reading both case roots without importing a full computer-algebra library.
FAQ
What happens if c is negative?
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 is not an equation in x—only a true or false statement about |b| and c. The solver requires a≠0 so it can return roots for x.
How many solutions does it return?
Negative c is rejected as no solution rather than an empty success list. Positive and negative case objects are always present.
What are the two cases of |2x − 3| = 5?
The positive case is 2x − 3 = 5, which gives 4. The negative case is 2x − 3 = −5, which gives −1. Both roots are shown with the equation they came from.
Which mistakes does writing both cases help avoid?
It stops you from dropping a sign, forgetting the second case or inventing a root when the right side is negative. Seeing each case next to its root makes every branch easy to check.
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/absolute-value-two-solutions \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":2,"b":-3,"c":5}'const res = await fetch("https://api.kit.forhosting.com/algebra/absolute-value-two-solutions", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 2,
"b": -3,
"c": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/absolute-value-two-solutions",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 2,
"b": -3,
"c": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/absolute-value-two-solutions", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":2,"b":-3,"c":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":2,"b":-3,"c":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/absolute-value-two-solutions", 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": -3,
"c": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.absolute_value_two_solutions",
"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
When c is strictly positive you get two distinct roots; when c is zero the cases collapse into one root; when c is negative the capability reports no solution through invalid_input instead of a misleading empty success list.
Why absolute value equations produce two candidate solutions
The capability encodes that impossibility as an invalid_input error rather than a success payload with zero solutions, matching classroom language that the equation has no solution and matching API clients that treat empty success and hard failure differently for billing and retries.
Input contract, two-case response shape, and error behaviour
Example: a equals two, b equals negative three, c equals five yields |2x-3|=5, positive case 2x-3 equals five with root four, negative case 2x-3 equals negative five with root negative one, and the ordered solutions list negative one then four. POST a JSON body with three required finite numbers: a, the non-zero coefficient of x inside the modulus; b, the constant term inside the modulus; and c, the non-negative right-hand side that the absolute value must equal. Each field may arrive as a JSON number or as a numeric string; non-finite values, missing fields, and empty strings fail with invalid_input naming the field. On success the body echoes a, b, and c; an equation string that rewrites the modulus form for display; positive_case and negative_case objects each carrying the linear equation string and its solved x; solution_count of one or two; solutions as an ascending array of numbers rounded to twelve decimal places for stable goldens across runtimes; and a formula string restating the two-case identity. Pair the response with your own graphing code if you need a sketch of the V-shaped graph y equals |ax+b| against the horizontal line y equals c; this endpoint solves for the two case roots only and does not emit plots or multi-paragraph step prose beyond the formula and case fields.
Where a two-solutions absolute value calculator earns its keep
Engineering and science notebooks often reduce piecewise-linear constraints or symmetric tolerance bands to |ax+b|=c when calibrating sensors; calling a tiny parse API that returns both case roots avoids pulling a full computer-algebra system for a three-coefficient problem.
Technical questions
How does the two-solutions calculator solve |ax + b| = c?
It splits into the positive case ax+b=c and the negative case ax+b=-c, solves each for x, and returns both roots ordered ascending when c>0 (or the single root when c=0).
What does the absolute value two solutions calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.