> ## Documentation Index
> Fetch the complete documentation index at: https://docs.iotools.cloud/llms.txt
> Use this file to discover all available pages before exploring further.

# Absolute Value Inequality Solver

> Solve an absolute value inequality of the form |ax + b| <, ≤, >, or ≥ c for x. Returns the solution in interval notation, handles the no-solution and all-reals edge cases, and shows a step-by-step derivation.

[Try Absolute Value Inequality Solver in your browser →](https://iotools.cloud/tool/absolute-value-inequality-solver/)



## OpenAPI

````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/absolute-value-inequality-solver
openapi: 3.1.0
info:
  title: iotools.cloud API
  version: 1.0.0
  description: >-
    Run any iotools.cloud tool over HTTP.


    Authenticate with `Authorization: Bearer iot_live_…`.


    **Only `POST /v1/tool/{slug}` costs credits.** Every GET here — the catalog,
    a tool's schema, your balance — is free. A tool call is charged its own
    weight or your plan's per-call minimum, whichever is larger;
    `x-iotools-credit-cost` on each operation is quoted at the free-tier
    minimum, and `GET /v1/tools/list` returns the exact figure for your key.
    `GET /v1/me/credits` reports what you have left, and `GET /v1/me/usage`
    reports what it went on.


    Failures are RFC 9457 problem documents — branch on `code`.
servers:
  - url: https://api.iotools.cloud
    description: Production
security:
  - bearerAuth: []
tags:
  - name: Catalog
    description: Find a tool and read its contract. Free.
  - name: Converters
    description: >-
      Convert between formats, encodings, and units — Base64, CSV and JSON,
      timestamps, and more. Fast, free, and processed right in your browser.
  - name: Formatters
    description: >-
      Format, beautify, minify, and validate code and data — JSON, HTML, CSS,
      SQL, and regex. Clean up messy input in one click, with nothing to upload.
  - name: Generators
    description: >-
      Generate exactly what you need and on demand — passwords, UUIDs, QR codes,
      hashes, random numbers, and more. Secure, instant, and free.
  - name: Calculators
    description: >-
      Crunch the numbers fast — from everyday math to specialized conversions
      and unit work. Free online calculators that run entirely in your browser.
  - name: Editors
    description: >-
      Edit and transform text, code, and images with quick, focused editors that
      run entirely in your browser — nothing to install, and no sign-up needed.
  - name: Utilities
    description: >-
      Everyday developer and web utilities — DNS and IP lookups, redirect and
      certificate checkers, and other quick diagnostics. Free and
      privacy-friendly.
  - name: Account
    description: Your key's allowance, limits and usage history.
paths:
  /v1/tool/absolute-value-inequality-solver:
    post:
      tags:
        - Calculators
      summary: Absolute Value Inequality Solver
      description: >-
        Solve an absolute value inequality of the form |ax + b| <, ≤, >, or ≥
        c for x. Returns the solution in interval notation, handles the
        no-solution and all-reals edge cases, and shows a step-by-step
        derivation.


        [Try Absolute Value Inequality Solver in your browser
        →](https://iotools.cloud/tool/absolute-value-inequality-solver/)
      operationId: run_absolute_value_inequality_solver
      requestBody:
        required: true
        content:
          application/json:
            schema:
              type: object
              properties:
                a:
                  type: number
                  description: a
                b:
                  type: number
                  description: b
                operator:
                  enum:
                    - lt
                    - le
                    - gt
                    - ge
                  description: Comparison
                c:
                  type: number
                  description: c
              required:
                - a
                - b
                - c
            examples:
              2x_3_5_bounded_interval:
                summary: '|2x - 3| ≤ 5 (bounded interval)'
                value:
                  a: '2'
                  b: '-3'
                  operator: le
                  c: '5'
              x_1_3_two_rays:
                summary: '|x + 1| > 3 (two rays)'
                value:
                  a: '1'
                  b: '1'
                  operator: gt
                  c: '3'
              x_1_no_solution_c_0:
                summary: '|x| < -1 (no solution, c < 0)'
                value:
                  a: '1'
                  b: '0'
                  operator: lt
                  c: '-1'
              x_2_all_reals_c_0:
                summary: '|x| ≥ -2 (all reals, c < 0)'
                value:
                  a: '1'
                  b: '0'
                  operator: ge
                  c: '-2'
              0x_5_5_a_0_no_solution:
                summary: '|0x + 5| < 5 (a = 0, no solution)'
                value:
                  a: '0'
                  b: '5'
                  operator: lt
                  c: '5'
              2x_4_0_single_point_c_0:
                summary: '|2x + 4| ≤ 0 (single point, c = 0)'
                value:
                  a: '2'
                  b: '4'
                  operator: le
                  c: '0'
              2x_4_0_all_reals_except_a_point_c_0:
                summary: '|2x + 4| > 0 (all reals except a point, c = 0)'
                value:
                  a: '2'
                  b: '4'
                  operator: gt
                  c: '0'
              2x_4_0_all_reals_non_strict_c_0:
                summary: '|2x + 4| ≥ 0 (all reals, non-strict, c = 0)'
                value:
                  a: '2'
                  b: '4'
                  operator: ge
                  c: '0'
      responses:
        '200':
          description: Tool output
          content:
            application/json:
              examples:
                2x_3_5_bounded_interval:
                  summary: '|2x - 3| ≤ 5 (bounded interval)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: bounded_interval
                      solution: '[-1, 4]'
                      steps: >-
                        Inequality: |2x + -3| ≤ 5

                        a ≠ 0 and c ≥ 0, so solve the two boundary equations: 2x
                        + -3 = 5 → x = 4, and 2x + -3 = -5 → x = -1

                        Boundaries sorted: lower = -1, upper = 4

                        |expr| ≤ 5 means the solution is the interval between
                        the boundaries: [-1, 4]
                    credits_used: 5
                    credits_remaining: null
                x_1_3_two_rays:
                  summary: '|x + 1| > 3 (two rays)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: two_rays
                      solution: (-∞, -4) ∪ (2, ∞)
                      steps: >-
                        Inequality: |1x + 1| > 3

                        a ≠ 0 and c ≥ 0, so solve the two boundary equations: 1x
                        + 1 = 3 → x = 2, and 1x + 1 = -3 → x = -4

                        Boundaries sorted: lower = -4, upper = 2

                        |expr| > 3 means the solution is everything outside the
                        boundaries: (-∞, -4) ∪ (2, ∞)
                    credits_used: 5
                    credits_remaining: null
                x_1_no_solution_c_0:
                  summary: '|x| < -1 (no solution, c < 0)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: no_solution
                      solution: >-
                        No solution (∅) — an absolute value can never be less
                        than a negative number.
                      steps: >-
                        Inequality: |1x + 0| < -1

                        c < 0 (-1 < 0). An absolute value is always ≥ 0, so it
                        can never be < a negative number.
                    credits_used: 5
                    credits_remaining: null
                x_2_all_reals_c_0:
                  summary: '|x| ≥ -2 (all reals, c < 0)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: all_reals
                      solution: >-
                        All real numbers (-∞, ∞) — an absolute value is always
                        greater than a negative number.
                      steps: >-
                        Inequality: |1x + 0| ≥ -2

                        c < 0 (-2 < 0). An absolute value is always ≥ 0, so it
                        can never be ≥ a negative number — the inequality holds
                        for every x.
                    credits_used: 5
                    credits_remaining: null
                0x_5_5_a_0_no_solution:
                  summary: '|0x + 5| < 5 (a = 0, no solution)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: no_solution
                      solution: No solution (∅)
                      steps: >-
                        Inequality: |0x + 5| < 5

                        a = 0, so the expression is constant: |5| = 5. Check 5 <
                        5: false.
                    credits_used: 5
                    credits_remaining: null
                2x_4_0_single_point_c_0:
                  summary: '|2x + 4| ≤ 0 (single point, c = 0)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: single_point
                      solution: '{-2}'
                      steps: >-
                        Inequality: |2x + 4| ≤ 0

                        a ≠ 0 and c ≥ 0, so solve the two boundary equations: 2x
                        + 4 = 0 → x = -2, and 2x + 4 = -0 → x = -2

                        Boundaries sorted: lower = -2, upper = -2

                        c = 0, so the "between the boundaries" interval
                        collapses to the single point x = -2.
                    credits_used: 5
                    credits_remaining: null
                2x_4_0_all_reals_except_a_point_c_0:
                  summary: '|2x + 4| > 0 (all reals except a point, c = 0)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: all_reals_except_point
                      solution: (-∞, -2) ∪ (-2, ∞)
                      steps: >-
                        Inequality: |2x + 4| > 0

                        a ≠ 0 and c ≥ 0, so solve the two boundary equations: 2x
                        + 4 = 0 → x = -2, and 2x + 4 = -0 → x = -2

                        Boundaries sorted: lower = -2, upper = -2

                        c = 0, so |expr| > 0 means the expression must be
                        nonzero — every real number except x = -2.
                    credits_used: 5
                    credits_remaining: null
                2x_4_0_all_reals_non_strict_c_0:
                  summary: '|2x + 4| ≥ 0 (all reals, non-strict, c = 0)'
                  value:
                    tool: absolute-value-inequality-solver
                    tool_version: 1.0.2
                    outputs:
                      status: all_reals
                      solution: All real numbers (-∞, ∞)
                      steps: >-
                        Inequality: |2x + 4| ≥ 0

                        a ≠ 0 and c ≥ 0, so solve the two boundary equations: 2x
                        + 4 = 0 → x = -2, and 2x + 4 = -0 → x = -2

                        Boundaries sorted: lower = -2, upper = -2

                        c = 0, so |expr| ≥ 0 holds everywhere, including exactly
                        at x = -2 where the expression is 0 — every real number.
                    credits_used: 5
                    credits_remaining: null
              schema:
                type: object
                required:
                  - tool
                  - tool_version
                  - outputs
                  - credits_used
                  - credits_remaining
                properties:
                  outputs:
                    type: object
                    properties:
                      solution:
                        type: string
                        description: Solution (interval notation)
                      status:
                        type: string
                        description: Status
                      steps:
                        type: string
                        description: Step-by-Step Derivation
                  tool:
                    type: string
                    description: The tool's slug, echoing the {slug} in the request path.
                  tool_version:
                    type: string
                    description: Output-contract version for this tool.
                  credits_used:
                    type: integer
                    description: >-
                      Credits this call consumed, after any settlement refund. 0
                      when metering is disabled.
                  credits_remaining:
                    type:
                      - integer
                      - 'null'
                    description: >-
                      Credits left in the current monthly allowance, or null
                      when metering is disabled.
                  request_id:
                    type: string
                    description: Correlation id, also sent as x-request-id.
        '400':
          $ref: '#/components/responses/ValidationError'
        '401':
          $ref: '#/components/responses/Unauthorized'
        '402':
          $ref: '#/components/responses/InsufficientCredits'
        '403':
          $ref: '#/components/responses/ToolNotAllowed'
        '404':
          $ref: '#/components/responses/ToolNotFound'
        '413':
          $ref: '#/components/responses/PayloadTooLarge'
        '422':
          $ref: '#/components/responses/UnprocessableInput'
        '429':
          $ref: '#/components/responses/RateLimited'
        '500':
          $ref: '#/components/responses/ToolFailed'
        '503':
          $ref: '#/components/responses/ToolDisabled'
      security:
        - bearerAuth: []
components:
  responses:
    ValidationError:
      description: Invalid request.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/validation_error
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Invalid request
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - validation_error
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
              fields:
                type:
                  - object
                  - string
                additionalProperties:
                  type: string
                description: >-
                  What failed: a map of field name → message. Absent when the
                  body itself is malformed; a single string when the failure
                  isn't tied to one field.
          example:
            type: https://docs.iotools.cloud/errors/validation_error
            title: Invalid request
            status: 400
            code: validation_error
            detail: One or more inputs are invalid — see `fields`.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
            fields:
              inputString: Required
    Unauthorized:
      description: Missing or invalid API key.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/invalid_api_key
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Invalid API key
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - invalid_api_key
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
          example:
            type: https://docs.iotools.cloud/errors/invalid_api_key
            title: Invalid API key
            status: 401
            code: invalid_api_key
            detail: 'Provide ''Authorization: Bearer <key>''.'
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
    InsufficientCredits:
      description: Monthly credit allowance exhausted.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/insufficient_credits
                  - >-
                    https://docs.iotools.cloud/errors/vendor_api_tier_limit_exceeded
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Insufficient credits
                  - Free-tier API limit reached for this tool
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - insufficient_credits
                  - vendor_api_tier_limit_exceeded
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
              credits_used:
                type: integer
                description: Always 0 — a refused call charges nothing.
              credits_remaining:
                type: integer
                description: Credits left in the allowance — fewer than this call costs.
          example:
            type: https://docs.iotools.cloud/errors/insufficient_credits
            title: Insufficient credits
            status: 402
            code: insufficient_credits
            detail: This call costs 1 credit and 0 remain in this month's allowance.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
            credits_used: 0
            credits_remaining: 0
    ToolNotAllowed:
      description: Tool exists but has no API surface.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/tool_not_allowed
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Tool not available over the API
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - tool_not_allowed
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
          example:
            type: https://docs.iotools.cloud/errors/tool_not_allowed
            title: Tool not available over the API
            status: 403
            code: tool_not_allowed
            detail: >-
              "Background Remover" is available on iotools.cloud but has no API
              endpoint.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
    ToolNotFound:
      description: No such tool.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/tool_not_found
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Tool not found
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - tool_not_found
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
          example:
            type: https://docs.iotools.cloud/errors/tool_not_found
            title: Tool not found
            status: 404
            code: tool_not_found
            detail: No tool with that slug. See GET /v1/tools/list.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
    PayloadTooLarge:
      description: Body too large.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/payload_too_large
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Payload too large
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - payload_too_large
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
          example:
            type: https://docs.iotools.cloud/errors/payload_too_large
            title: Payload too large
            status: 413
            code: payload_too_large
            detail: Request body exceeds this tool's size limit.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
    UnprocessableInput:
      description: Well-formed request, but there is no result to return.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/unprocessable_input
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Input cannot be processed
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - unprocessable_input
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
              credits_used:
                type: integer
                description: >-
                  Always 0 on `unprocessable_input` — a failed run is refunded,
                  floor included.
              credits_remaining:
                type: integer
                description: >-
                  Credits left after the refund. Absent when metering is
                  disabled.
          example:
            type: https://docs.iotools.cloud/errors/unprocessable_input
            title: Input cannot be processed
            status: 422
            code: unprocessable_input
            detail: >-
              This video has no captions or subtitles, so there is no transcript
              to extract.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
            credits_used: 0
    RateLimited:
      description: Per-minute rate limit exceeded.
      headers:
        Retry-After:
          description: Seconds to wait before retrying (RFC 9110 delta-seconds).
          schema:
            type: integer
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/rate_limited
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Rate limit exceeded
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - rate_limited
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
              retry_after:
                type: integer
                description: >-
                  Seconds until the window resets — the same value as the
                  `Retry-After` header.
          example:
            type: https://docs.iotools.cloud/errors/rate_limited
            title: Rate limit exceeded
            status: 429
            code: rate_limited
            detail: Too many requests. Retry in 30s.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
            retry_after: 30
    ToolFailed:
      description: Tool failed to run.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/processing_error
                  - https://docs.iotools.cloud/errors/internal_error
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Tool failed to run
                  - Internal error
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - processing_error
                  - internal_error
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
              credits_used:
                type: integer
                description: >-
                  Always 0 on `processing_error` — a failed run is refunded,
                  floor included.
              credits_remaining:
                type: integer
                description: >-
                  Credits left after the refund. Absent when metering is
                  disabled.
          example:
            type: https://docs.iotools.cloud/errors/processing_error
            title: Tool failed to run
            status: 500
            code: processing_error
            detail: The tool failed to run. Please try again.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
            credits_used: 0
    ToolDisabled:
      description: Tool temporarily disabled.
      content:
        application/problem+json:
          schema:
            type: object
            description: >-
              RFC 9457 problem document, served as application/problem+json.
              Branch on `code`; `title` is human prose and may be reworded
              without notice. Some failures add extension members — `fields` on
              validation errors, `retry_after` on 429s,
              `credits_used`/`credits_remaining` on billing-adjacent failures —
              documented on the responses that carry them.
            required:
              - type
              - title
              - status
              - code
            properties:
              type:
                type: string
                format: uri
                description: Stable documentation URI for this failure.
                examples:
                  - https://docs.iotools.cloud/errors/tool_disabled
                  - https://docs.iotools.cloud/errors/api_unconfigured
              title:
                type: string
                description: Short human-readable summary of the failure.
                examples:
                  - Tool temporarily disabled
                  - API not configured
              status:
                type: integer
                description: HTTP status code, matching the response's own status.
              code:
                type: string
                description: >-
                  Stable machine-readable error code — branch on this, not
                  `title`.
                enum:
                  - tool_disabled
                  - api_unconfigured
              detail:
                type: string
                description: Human explanation of this occurrence.
              request_id:
                type: string
                description: Correlation id, also sent as x-request-id.
          example:
            type: https://docs.iotools.cloud/errors/tool_disabled
            title: Tool temporarily disabled
            status: 503
            code: tool_disabled
            detail: This tool is temporarily unavailable. Try again shortly.
            request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
  securitySchemes:
    bearerAuth:
      type: http
      scheme: bearer
      bearerFormat: iot_live_…

````