Solve and check. ( 3 marks)
5/(x-1) + 2/(x+1)=-6
Here's my work on it:
5/(x-1)+2/(x+1)=?
5/(-3/2-1)+2/(-3/2+1)=-6
Can you show how I would have found x in this situation and also if I went wrong anywhere.
[5(x+1) +2(x-1)]/(x^2-1) = -6
5x+5 +2x-2 = -6x^2 + 6
6 x^2 + 7 x - 3 = 0
(2x+3)(3x-1) = 0
x = 1/3 or -3/2
The first thing I would do is get rid of the fractions
The LCD is (x-1)(x+1) so let's multiply each term by that ...
5(x+1) + 2(x-1) = -6(x+1)(x-1)
5x + 5 + 2x - 2 = -6x^2 + 6
6x^2 + 7x - 3 = 0
(3x - 1)(2x + 3) = 0
x = 1/3 or x = -3/2
I won't even attempt to figure out what you were trying to do in your last line.
To solve the equation 5/(x-1) + 2/(x+1) = -6, follow these steps:
Step 1: Common Denominator
To simplify the equation, find a common denominator for the fractions. In this case, the common denominator is (x-1)(x+1) because it includes both (x-1) and (x+1). Therefore, rewrite the equation with the common denominator:
[(5(x+1))/(x-1)(x+1)] + [(2(x-1))/(x-1)(x+1)] = -6
Step 2: Add the fractions
Combine the fractions together by adding the numerators, while keeping the common denominator:
(5(x+1) + 2(x-1))/(x-1)(x+1) = -6
Simplifying further:
(5x + 5 + 2x - 2)/(x-1)(x+1) = -6
(7x + 3)/(x-1)(x+1) = -6
Step 3: Cross Multiply
Remove the fraction by cross multiplying:
(7x + 3) = -6(x-1)(x+1)
Step 4: Distribute and simplify
Distribute -6 to both (x-1) and (x+1):
7x + 3 = -6(x^2 - 1)
7x + 3 = -6x^2 + 6
Combine like terms:
0 = 6x^2 + 7x - 3
Step 5: Solve the quadratic equation
To solve the quadratic equation, you can either factor it, complete the square, or use the quadratic formula. In this case, factoring is not straightforward. So, let's use the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / (2a)
a = 6, b = 7, 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
We get two possible solutions:
x = (-7 + 11) / 12 = 4/12 = 1/3
x = (-7 - 11) / 12 = -18/12 = -3/2
Step 6: Check the solutions
Verify if the solutions satisfy the original equation. Substitute each value back into the original equation and check if both sides are equal.
For x = 1/3:
5/(1/3 - 1) + 2/(1/3 + 1) = -6
5/(-2/3) + 2/(4/3) = -6
(-15/2) + 3 = -6
-15/2 + 6/2 = -6
-9/2 = -6
The left side is not equal to the right side, so x = 1/3 is not a solution.
For x = -3/2:
5/(-3/2 - 1) + 2/(-3/2 + 1) = -6
5/(-3/2 - 2/2) + 2/(-3/2 + 2/2) = -6
5/(-5/2) + 2/(-1/2) = -6
(-10/5) + (-4/1) = -6
-2 - 4 = -6
-6 = -6
The left side is equal to the right side, so x = -3/2 is a solution.
Therefore, the solution to the equation 5/(x-1) + 2/(x+1) = -6 is x = -3/2 only.