Integrals of absolute values
Problem 4.589 · easy
Evaluate \( \displaystyle \int_{-4}^{-1} \left| 3 x + 9 \right| dx \).
- \[ 3 x + 9 \]The integrand is zero at x = -3.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{-4}^{-3} \left(- 3 x - 9\right)\, dx + \int\limits_{-3}^{-1} \left(3 x + 9\right)\, dx = \frac{15}{2} \]Integrate each piece and add.✓ Proved
Answer \( \frac{15}{2} \)
Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature of |p(x)| |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the root of the linear term, splits the integral at that point, and applies the correct definition of the absolute value for each sub-interval. The final calculation is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the root of the linear term, splits the integral at that point, and applies the correct definition of the absolute value for each sub-interval. The final calculation is correct.gpt-oss:20b: fail (error) 2026-10-07 — The sentence claims that “between those zeros the sign does not change,” yet the function 3x+9 has only one zero at x=−3. On the interval [−4,−1] the sign actually changes at x=−3, so the statement about the sign is incorrect and would mislead a student.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the root of the linear function, splits the integral at that point, and applies the correct sign for the absolute value in each sub-interval. The final calculation is correct.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/absolute_value_integral, checked 2026-10-07 with SymPy 1.14.0.