Solving fractional equations
1)
(1/x-3)+(1/x+3)=(10/x^2-9)
2)
(5/2x)-(5/3(x+5)=(5/x)
3)
(2x-1/x-1)-(3x/2x+1)=(x^2+11/2x^2-x-1)
I just have a problem knowing what the lcd is. For the first one, i know I had to m,multiply the left side by x-3 and x+3, but it had to be the same and you had to multiply all of them by it, so I kind of got lost. Could someone just help me get started for each one and form there I should be able to figure it out.
x^2-9 = (x+3)(x-3)
so that is the lcd
1/(x-3) + 1/(x+3) = 10/(x^2-9)
multiply through by x^2-9 and you have
(x+3) + (x-3) = 10
2x = 10
x = 5
For #2 if you mean
5/(2x) - 5/(3(x+5)) = (5/x)
the lcd is 2x(3(x+5)) so
5(3(x+5) - 5(2x) = 5(2)(3(x+5))
15x+75 - 10x = 30x+150
If I got the parentheses wrong, I'm sure you can fix it up.
For #3 I assume
(2x-1)/(x-1)-3x/(2x+1)=(x^2+11)/(2x^2-x-1)
the lcd is (x-1)(2x+1) = 2x^2-x-1, so
(2x-1)(2x+1) - 3x(x-1) = (x^2+11)
4x^2-1-3x^2+3x = x^2+11
...
OHHHH, wait, correct me if i'm wrong, but the lcd for problems like this are most likely the denominator of the 2 fraction being combined?
yes, just as with normal fractions
2/3 + 4/7
has an lcd of 4*7
3/4 + 1/6
has an lcd of 12 instead of 24, since 2 is a common factor of both 4 and 6.
LCD(m,n) = mn/GCF(mn)
Thank you so much
Sure! I can help you get started with each of the fractional equations and explain the process of solving them.
1) To solve the equation (1/x-3) + (1/x+3) = (10/x^2-9), we need to find a common denominator or least common denominator (LCD) for the fractions. The LCD is the smallest multiple of the denominators, in this case, x-3, x+3, and (x^2-9). Let's break down how to find the LCD step by step:
Step 1: Factor the denominators.
The denominator x^2-9 can be factored as (x-3)(x+3).
Step 2: Identify the factors.
The factors in the denominators are (x-3), (x+3), and (x-3)(x+3).
Step 3: Determine the LCD.
The LCD is the product of all the unique factors. In this case, the LCD is (x-3)(x+3).
Now, we can multiply both sides of the equation by the LCD to eliminate the fractions. Multiply each term on both sides by (x-3)(x+3):
(x-3)(x+3) * (1/x-3) + (x-3)(x+3) * (1/x+3) = (x-3)(x+3) * (10/x^2-9)
Simplifying the equation using the distributive property:
(x+3) + (x-3) = 10
Combine like terms:
2x = 10
Divide both sides by 2:
x = 5
So, the solution to the equation is x = 5.
Now, let's move on to the next equation.
2) The equation is (5/2x) - (5/3(x+5)) = (5/x). Similarly, we need to find the LCD for this equation.
Step 1: Factor the denominators.
No factoring is required here.
Step 2: Identify the factors.
The factors in the denominators are (2x), (3), and (x+5).
Step 3: Determine the LCD.
The LCD is the product of all the unique factors. In this case, the LCD is 2x * 3 * (x+5).
Now, multiply both sides of the equation by the LCD:
(2x * 3 * (x+5)) * (5/2x) - (2x * 3 * (x+5)) * (5/3(x+5)) = (2x * 3 * (x+5)) * (5/x)
Simplifying the equation:
15(x+5) - 10x * (x+5) = 30x
Expanding and combining like terms:
15x + 75 - 10x^2 - 50x = 30x
Rearranging the terms:
-10x^2 + 15x - 80x + 75 - 30x = 0
Combining like terms:
-10x^2 - 95x + 75 = 0
Now, you can use the quadratic formula or any other method to solve this quadratic equation. The solutions will give you the values of x.
Let's move on to the last equation.
3) The equation is (2x-1/x-1) - (3x/2x+1) = (x^2+11/2x^2-x-1).
Step 1: Factor the denominators.
No factorization is required for this equation.
Step 2: Identify the factors.
The factors in the denominators are (x-1) and (2x+1).
Step 3: Determine the LCD.
The LCD is the product of all the unique factors. In this case, the LCD is (x-1)(2x+1).
Multiply both sides of the equation by the LCD:
(x-1)(2x+1) * (2x-1/x-1) - (x-1)(2x+1) * (3x/2x+1) = (x-1)(2x+1) * (x^2+11/2x^2-x-1)
Simplifying the equation:
(2x-1) - 3x * (x-1) = x^2 + 11
Expanding and combining like terms:
2x - 1 - 3x^2 + 3x = x^2 + 11
Rearranging the terms:
-3x^2 - x^2 + 2x + 3x + 1 = 11
Combining like terms:
-4x^2 + 5x + 1 - 11 = 0
Simplifying:
-4x^2 + 5x - 10 = 0
Now, you can use the quadratic formula or any other method to solve this quadratic equation. The solutions will give you the values of x.
I hope this helps you understand how to get started with solving fractional equations and find their LCD.