1/x+1-1/x+2=1/6
Must have 2 solutions
1 / x + 1 - 1 / x + 2 = 1 / 6
1 / x - 1 / x + 1 + 2 = 1 / 6
3 = 1 / 6
Makes no sense.
however,
1/(x+1) - 1/(x+2) = 1/6
can be solved by rewriting it as
6(x+2) - 6(x+1) = (x+1)(x+2)
6x + 12 - 6x - 6 = x^2 + 3x + 2
x^2 + 3x - 4 = 0
(x+4)(x-1) = 0
x = -4,1
Now you know how useful parentheses can be. Always use them when posting problems where it's not clear how to group the terms.
To solve the equation 1/(x+1) - 1/(x+2) = 1/6 and find the two solutions, we can follow these steps:
Step 1: Create a common denominator
Multiply every term in the equation by a common denominator that is a multiple of (x+1) and (x+2). In this case, we can multiply every term by 6(x+1)(x+2) to eliminate the denominators. This will give us:
6(x+2) - 6(x+1) = (x+1)(x+2)
Step 2: Expand and simplify
Distribute 6 to (x+2) and (x+1) on the left side of the equation:
6x + 12 - 6x - 6 = x^2 + 3x + 2
Combine like terms:
6 - 6 = x^2 + 3x + 2
0 = x^2 + 3x + 2
Step 3: Solve the quadratic equation
Set the equation equal to zero by moving all terms to one side:
x^2 + 3x + 2 = 0
To solve the quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, the equation can be easily factored as:
(x+1)(x+2) = 0
Setting each factor equal to zero, we get two equations:
x+1 = 0 or x+2 = 0
Solving for x in each equation, we find the two solutions:
x = -1 or x = -2
Therefore, the equation 1/(x+1) - 1/(x+2) = 1/6 has two solutions: x = -1 and x = -2.