To solve the equation \(|\csc x| = \frac{5\pi}{4} - \left|\frac{x}{2}\right|\) for \(x\) in the interval \((-2\pi, 2\pi)\), we will follow these steps:
### Step 1: Define the Functions
Let:
- \(f(x) = |\csc x|\)
- \(g(x) = \frac{5\pi}{4} - \left|\frac{x}{2}\right|\)
### Step 2: Analyze the Function \(f(x)\)
The function \(f(x) = |\csc x|\) is defined as:
\[
|\csc x| = \frac{1}{|\sin x|}
\]
This function has vertical asymptotes where \(\sin x = 0\), which occurs at \(x = n\pi\) for \(n \in \mathbb{Z}\). In the interval \((-2\pi, 2\pi)\), the points where \(\sin x = 0\) are:
- \(x = -2\pi\)
- \(x = -\pi\)
- \(x = 0\)
- \(x = \pi\)
- \(x = 2\pi\)
### Step 3: Analyze the Function \(g(x)\)
The function \(g(x) = \frac{5\pi}{4} - \left|\frac{x}{2}\right|\) is a linear function with a slope of \(-\frac{1}{2}\). The y-intercept is \(\frac{5\pi}{4}\).
### Step 4: Find Intersections
To find the number of solutions, we need to determine where the graphs of \(f(x)\) and \(g(x)\) intersect.
1. **Behavior of \(f(x)\)**:
- As \(x\) approaches \(n\pi\) from the left or right, \(f(x)\) approaches infinity.
- Between the asymptotes, \(f(x)\) decreases from infinity to 1 and then increases back to infinity.
2. **Behavior of \(g(x)\)**:
- The function \(g(x)\) is a straight line that decreases as \(x\) increases.
- The maximum value occurs at \(x = 0\), where \(g(0) = \frac{5\pi}{4}\).
### Step 5: Determine Points of Intersection
- In each interval \((-2\pi, -\pi)\), \((- \pi, 0)\), \((0, \pi)\), and \((\pi, 2\pi)\), the function \(f(x)\) will intersect the line \(g(x)\) twice (once when decreasing and once when increasing).
- Therefore, we can expect 2 intersections in each of the 4 intervals.
### Step 6: Count the Total Solutions
Since there are 4 intervals and each has 2 intersections, the total number of solutions is:
\[
4 \text{ intervals} \times 2 \text{ intersections per interval} = 8 \text{ solutions}
\]
### Conclusion
The total number of solutions to the equation \(|\csc x| = \frac{5\pi}{4} - \left|\frac{x}{2}\right|\) in the interval \((-2\pi, 2\pi)\) is **8**.
