Question

Solve and Check

5/x-1 + 2/x+1 = -6

Answer 1

To solve this equation, we first need to find a common denominator for the fractions on the left side:

5/(x-1) + 2/(x+1) = -6

Multiplying the first fraction by (x+1) and the second fraction by (x-1):

5(x+1)/(x-1)(x+1) + 2(x-1)/(x-1)(x+1) = -6

Now our equation becomes:

5(x+1) + 2(x-1) = -6(x-1)(x+1)

Expanding both sides:

5x + 5 + 2x - 2 = -6(x^2 - 1)

Combining like terms:

7x + 3 = -6x^2 + 6

Rearranging the equation to set it equal to 0:

6x^2 + 7x - 3 = 0

Now we need to solve this quadratic equation. We can do this using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 6, b = 7, and c = -3. Plugging these values into the formula:

x = (-(7) ± √(7^2 - 4*6*(-3))) / 2*6
x = (-7 ± √(49 + 72)) / 12
x = (-7 ± √121) / 12
x = (-7 ± 11) / 12

This gives two possible solutions:

x = (4) / 12 = 1/3
x = (-18) / 12 = -3/2

Now let's check if these solutions are correct by substituting back into the original equation:

For x = 1/3:

5/(1/3-1) + 2/(1/3+1) = 5/(-2/3) + 2/(4/3)
= -15/2 + 6/4
= -15/2 + 3/2
= -12/2
= -6

For x = -3/2:

5/(-3/2-1) + 2/(-3/2+1) = 5/(-5/2) + 2/(-1/2)
= -10/5 + 4/-1
= -2 + 4/-1
= -2 - 4
= -6

Both solutions check out, so x = 1/3 and x = -3/2 are valid solutions to the equation.