Solution:

Introduction:
To find the number of real solutions of the given equation x - 1/x^2 - 4 = 2 - 1/x^2 - 4, we will solve the equation and determine the number of real solutions.

Step 1: Simplify the equation
We can simplify the given equation as follows:

x - 1/(x^2 - 4) = 2 - 1/(x^2 - 4)

Multiplying both sides by (x^2 - 4), we get

x(x^2 - 4) - 1 = 2(x^2 - 4) - 1

Expanding and simplifying, we get

x^3 - 4x - 7 = 0

Step 2: Use the Rational Root Theorem
To find the real solutions of the equation x^3 - 4x - 7 = 0, we can use the Rational Root Theorem. This theorem states that any rational root of the equation must be of the form p/q, where p is a factor of the constant term (-7 in this case) and q is a factor of the leading coefficient (1 in this case).

The factors of -7 are ±1, ±7, so the possible rational roots are:

±1, ±7, ±1/2, ±7/2

We can check each of these values to see if they are solutions of the equation.

Step 3: Check the rational roots
Checking x = 1, we get

1^3 - 4(1) - 7 = -10

Checking x = -1, we get

(-1)^3 - 4(-1) - 7 = 2

Checking x = 7, we get

7^3 - 4(7) - 7 = 314

Checking x = -7, we get

(-7)^3 - 4(-7) - 7 = -252

Checking x = 1/2, we get

(1/2)^3 - 4(1/2) - 7 = -21/8

Checking x = -1/2, we get

(-1/2)^3 - 4(-1/2) - 7 = -25/8

Checking x = 7/2, we get

(7/2)^3 - 4(7/2) - 7 = 189/8

Checking x = -7/2, we get

(-7/2)^3 - 4(-7/2) - 7 = -203/8

Step 4: Determine the number of real solutions
From the above calculations, we can see that none of the rational roots are solutions of the equation x^3 - 4x - 7 = 0. Therefore, the equation has no rational solutions.

However, there may be irrational or complex solutions. To determine the number of real solutions, we can use the Intermediate Value Theorem. This theorem states that if f(x) is a continuous function on an interval [a, b], and if f(a) and f(b) have opposite signs, then there must be at least one real root of the equation f(x) = 0 in

There is no solution for your given equation
The denominator is not defined at x=2,-2
But 1st term is x=2
So no real solution