Question

What is the solution to the equation x+3/x+2=3+1/x

Answer 1

The solution set is { -1 }.

Explanation:

If we multiply each term by the LCM of x and x+ 2 ( = x(x + 2)) we get:-

x(x + 3) = 3x(x + 2) + x + 2

x^2 + 3x = 3x^2 + 6x + x + 2

0 = 3x^2 - x^2 - 3x + 6x + x + 2

2x^2 +4x + 2 = 0

2(x^2 + 2x + 1) = 0

2( x + 1)(x + 1) = 0

x = -1 Multiplicity 2

The solution set only contains 1 * -1 . Elements in a set are not repeated

So its {-1} (answer)

Answer 2

x=-1

Explanation:

x+3

------- = 3 + 1/x

x+2

Get a common denominator on the right

3*x/x + 1/x

3x/x + 1/x = (3x+1)/x

x+3 3x+1

------- = ---------

x+2 x

We can use cross products to solve

(x+3) *x = (3x+1) * (x+2)

x^2+3x = 3x*x +x + 3x*2 +2

Simplifying

x^2 + 3x = 3x^2 +7x+2

Subtract x^2 from each side

x^2 -x^2 + 3x = 3x^2-x^2 +7x+2

3x = 2x^2 +7x+2

Subtract 3x from each side

3x-3x = 2x^2 +7x -3x+2

0 = 2x^2 +4x +2

Divide by 2

0 = x^2 +2x +1

Factor

0 = (x+1) (x+1)

Using the zero product property

x+1 =0

x=-1