Find the number of roots of the equation `7^(|x|)(|5-|x||)=1`

Answer

To find the number of roots of the equation \( 7^{|x|}(|5 - |x||) = 1 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 7^{|x|}(|5 - |x||) = 1 \]
This can be rewritten as:
\[ 7^{|x|} = \frac{1}{|5 - |x||} \]

### Step 2: Analyze the function on both sides
The left side, \( 7^{|x|} \), is an exponential function that is always positive and increases as \( |x| \) increases. The right side, \( \frac{1}{|5 - |x||} \), is a rational function that is also positive but has vertical asymptotes at \( |x| = 5 \).

### Step 3: Determine critical points
The critical points occur when \( |x| = 5 \) because this is where the right side becomes undefined (infinity). We need to analyze the behavior of both sides of the equation around these points.

### Step 4: Evaluate at specific points
1. **At \( |x| = 0 \)**:
\[ 7^{|0|} = 7^0 = 1 \]
\[ |5 - |0|| = 5 \implies \frac{1}{5} \]
So, \( 1 \neq \frac{1}{5} \).

2. **At \( |x| = 5 \)**:
The right side becomes undefined (infinity).

3. **At \( |x| = 1 \)**:
\[ 7^{|1|} = 7^1 = 7 \]
\[ |5 - |1|| = 4 \implies \frac{1}{4} \]
So, \( 7 \neq \frac{1}{4} \).

4. **At \( |x| = 4 \)**:
\[ 7^{|4|} = 7^4 \]
\[ |5 - |4|| = 1 \implies \frac{1}{1} = 1 \]
So, \( 7^4 \neq 1 \).

5. **At \( |x| = 2 \)**:
\[ 7^{|2|} = 7^2 = 49 \]
\[ |5 - |2|| = 3 \implies \frac{1}{3} \]
So, \( 49 \neq \frac{1}{3} \).

### Step 5: Sketch the graphs
Now we can sketch the graphs of \( y = 7^{|x|} \) and \( y = \frac{1}{|5 - |x||} \).

- The graph of \( y = 7^{|x|} \) is an increasing function that starts at \( (0, 1) \) and goes to infinity as \( |x| \) increases.
- The graph of \( y = \frac{1}{|5 - |x||} \) approaches infinity at \( |x| = 5 \) and decreases towards zero as \( |x| \) moves away from 5.

### Step 6: Find intersections
From the sketch, we can see that there are four points where the two graphs intersect:
1. One point in the interval \( (-5, 0) \)
2. One point in the interval \( (0, 5) \)
3. One point in the interval \( (-5, -4) \)
4. One point in the interval \( (4, 5) \)

### Conclusion
Thus, the total number of roots of the equation \( 7^{|x|}(|5 - |x||) = 1 \) is **4**.

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