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The question appears to involve proving the inequality related to definite integrals:

\[
\left| \int_a^b f(x) \, dx \right| \leq \int_a^b |f(x)| \, dx,
\]

where \( f(x) \) and \( |f(x)| \) are integrable over the interval \([a, b]\).

Here is the step-by-step proof:

---

### Proof:
1. **Absolute Value of an Integral:**
   The absolute value of the integral of \(f(x)\) over \([a, b]\) can be expressed as:
   \[
   \left| \int_a^b f(x) \, dx \right|.
   \]

2. **Property of Absolute Values:**
   For any real number \(c\), \(|c| \geq c\) and \(|c| \geq -c\). Applying this property to the integrand \(f(x)\), we have:
   \[
   |f(x)| \geq f(x) \quad \text{and} \quad |f(x)| \geq -f(x).
   \]

3. **Comparison of Integrals:**
   Since \( |f(x)| \geq f(x) \), integrating over \([a, b]\) gives:
   \[
   \int_a^b |f(x)| \, dx \geq \int_a^b f(x) \, dx.
   \]
   Similarly, since \( |f(x)| \geq -f(x) \), integrating gives:
   \[
   \int_a^b |f(x)| \, dx \geq -\int_a^b f(x) \, dx.
   \]

4. **Combining Results:**
   These inequalities imply that:
   \[
   -\int_a^b |f(x)| \, dx \leq \int_a^b f(x) \, dx \leq \int_a^b |f(x)| \, dx.
   \]

5. **Taking the Absolute Value:**
   The absolute value of \(\int_a^b f(x) \, dx\) satisfies:
   \[
   \left| \int_a^b f(x) \, dx \right| \leq \int_a^b |f(x)| \, dx.
   \]

---

### Conclusion:
The inequality is proven:
\[
\left| \int_a^b f(x) \, dx \right| \leq \int_a^b |f(x)| \, dx.
\]

Let me know if you would like a further explanation or examples!

---

### Related Questions:
1. What is the significance of the triangle inequality in this proof?
2. How does this inequality relate to the properties of integrals over symmetric intervals?
3. Can the equality hold? If so, under what conditions?
4. How would this inequality apply if \(f(x)\) is piecewise continuous?
5. How is this inequality extended to improper integrals?

### Tip:
Always use the properties of absolute values and inequalities carefully when working with integrals. They help simplify and generalize complex expressions!

Prove that |∫_a^b f(x) dx| ≤ ∫_a^b |f(x)| dx when f(x) and |f(x)| are integrable on the interval [a, b].

Explore a detailed proof of the inequality |∫_a^b f(x) dx| ≤ ∫_a^b |f(x)| dx, using properties of absolute value and definite integrals. Ideal for calculus students in Grades 10-12 or college-level courses.

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