Equation

Simplifying each side of the equation results in [tex]\( x^2 - 3x - 4 = x^2 - 5x + 6 \)[/tex].

Find the solution:

[tex]\[
\frac{x+2}{3x} - \frac{1}{x-2} = \frac{x-3}{3x}
\][/tex]

[tex]\( x = \)[/tex]

[tex]\(\boxed{\phantom{x}}\)[/tex]



Answer :

To solve the equation:

[tex]\[ \frac{x+2}{3x} - \frac{1}{x-2} = \frac{x-3}{3x} \][/tex]

we start by simplifying each side of the equation.

1. Combine the terms on the left side into a single fraction:

[tex]\[ \frac{x+2}{3x} - \frac{1}{x-2} \][/tex]

To combine these fractions, we need a common denominator. The common denominator for [tex]\( \frac{x+2}{3x} \)[/tex] and [tex]\( \frac{1}{x-2} \)[/tex] is [tex]\( 3x(x-2) \)[/tex].

- Adjust the fractions to have a common denominator:

[tex]\[ \frac{(x+2)(x-2)}{3x(x-2)} - \frac{3x}{3x(x-2)} \][/tex]

Notice that [tex]\( \frac{x-3}{3x} \)[/tex] on the right already has the denominator [tex]\( 3x \)[/tex], we will write it out for simplification.

2. Simplify the numerators:

- For the first term [tex]\( \frac{x+2}{3x} \)[/tex]:

[tex]\[ \frac{(x+2)(x-2)}{3x(x-2)} = \frac{x^2 - 4}{3x(x-2)} \][/tex]

- For the second term [tex]\( \frac{3}{3x} \)[/tex]:

[tex]\[ \frac{3}{x-2} = \frac{3x}{3x(x-2)} \][/tex]

So the left-hand side rewrites as:

[tex]\[ \frac{x^2 - 4 - 3x}{3x(x-2)} \][/tex]

3. Combining and simplifying the expressions:

Combine the terms over the common denominator:

[tex]\[ \frac{x^2 - 4 - 3x}{3x(x-2)} = \frac{x^2 - 3x - 4}{3x(x-2)} \][/tex]

Likewise, the right-hand side is:

[tex]\[ \frac{x-3}{3x} \][/tex]

Thus, we have:

[tex]\[ \frac{x^2 - 3x - 4}{3x(x-2)} = \frac{x-3}{3x} \][/tex]

Multiplying both sides of the equation by [tex]\( 3x(x-2) \)[/tex] to eliminate the fractions:

[tex]\[ x^2 - 3x - 4 = (x-3)(x-2) \][/tex]

4. Expand and simplify the equation:

Expand the right-hand side:

[tex]\[ x^2 - 3x - 4 = x^2 - 5x + 6 \][/tex]

Subtract [tex]\( x^2 \)[/tex] from both sides:

[tex]\[ -3x - 4 = -5x + 6 \][/tex]

Isolate [tex]\( x \)[/tex]:

First, add [tex]\( 5x \)[/tex] to both sides:

[tex]\[ 2x - 4 = 6 \][/tex]

Then add 4 to both sides:

[tex]\[ 2x = 10 \][/tex]

Finally, divide both sides by 2:

[tex]\[ x = 5 \][/tex]

Therefore, the solution to the equation is:

[tex]\[ \boxed{5} \][/tex]