To solve the inequality \( \left| \sin x + \cos x \right| > 0 \), follow these steps:

### Step 1: Analyze the inequality
The inequality involves the absolute value of the expression \( \sin x + \cos x \), so we are looking for the values of \( x \) where \( \sin x + \cos x \neq 0 \), because the absolute value of a nonzero number is always greater than 0.

### Step 2: Solve the equation \( \sin x + \cos x = 0 \)
Let's solve for \( x \) where \( \sin x + \cos x = 0 \).

\[
\sin x + \cos x = 0
\]

Divide both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)):

\[
\tan x = -1
\]

This equation is satisfied when:

\[
x = \frac{3\pi}{4} + n\pi \quad \text{for some integer } n
\]

These are the values of \( x \) where \( \sin x + \cos x = 0 \).

### Step 3: Express the solution
The absolute value inequality \( \left| \sin x + \cos x \right| > 0 \) is satisfied for all \( x \) except when \( \sin x + \cos x = 0 \).

Thus, the solution to the inequality is:

\[
x \neq \frac{3\pi}{4} + n\pi \quad \text{for some integer } n
\]

### Conclusion:
The inequality \( \left| \sin x + \cos x \right| > 0 \) is satisfied for all \( x \) except for \( x = \frac{3\pi}{4} + n\pi \), where \( n \) is an integer.

Would you like more details on this, or have any further questions?

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Here are 5 related questions to extend your understanding:
1. How would you solve \( \left| \sin x - \cos x \right| > 0 \)?
2. What is the solution to \( \sin x + \cos x = 1 \)?
3. Can you express \( \sin x + \cos x \) as a single trigonometric function?
4. What is the range of \( \sin x + \cos x \)?
5. How does this inequality change if we have \( \left| \sin x + \cos x \right| \geq 1 \)?

**Tip**: When working with absolute value inequalities, remember to first solve for where the expression inside the absolute value equals zero, then check intervals for solutions!

This page explains how to solve the trigonometric inequality |sin x + cos x| > 0. Follow a step-by-step guide to find the values of x where sin x + cos x is nonzero. Includes key trigonometric concepts, formulas, and solutions to help students in Grades 10-12 understand absolute value inequalities in trigonometry.

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