The number of integral solution of the equation `|x^(2)-7|le 9` are

Answer

To solve the equation \( |x^2 - 7| \leq 9 \) and find the number of integral solutions, we can follow these steps:

### Step 1: Rewrite the absolute value inequality
The inequality \( |x^2 - 7| \leq 9 \) can be rewritten as two separate inequalities:
\[ -9 \leq x^2 - 7 \leq 9 \]

### Step 2: Break it into two inequalities
From the rewritten inequality, we can break it into two parts:
1. \( x^2 - 7 \geq -9 \)
2. \( x^2 - 7 \leq 9 \)

### Step 3: Solve the first inequality
For the first inequality \( x^2 - 7 \geq -9 \):
\[ x^2 \geq -9 + 7 \]
\[ x^2 \geq -2 \]
Since \( x^2 \) is always non-negative, this inequality is always satisfied for all real numbers \( x \).

### Step 4: Solve the second inequality
Now, for the second inequality \( x^2 - 7 \leq 9 \):
\[ x^2 \leq 9 + 7 \]
\[ x^2 \leq 16 \]
Taking the square root of both sides, we get:
\[ -4 \leq x \leq 4 \]

### Step 5: Identify integral solutions
The integral solutions for \( x \) are the integers that lie within the interval \( [-4, 4] \). The integers in this range are:
\[ -4, -3, -2, -1, 0, 1, 2, 3, 4 \]

### Step 6: Count the integral solutions
Counting these values, we find there are a total of 9 integral solutions.

### Final Answer
The number of integral solutions of the equation \( |x^2 - 7| \leq 9 \) is \( 9 \).

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