1/x+1/x+5+1/3x=1/2.25
Solve for x.
do you mean
1/x + 1/(x+5) + 1/(3x) = 100/225 ????
225/x +225/(x+5) + 75/(x) = 100
300/x + 225/(x+5) = 100
(x+5)(300) + 225 x = 100x(x+5)
300 x + 1500 + 225 x = 100 x^2 + 500 x
100 x^2 -25 x -1500 = 0
20 x^2 - 5 x - 300 = 0
4 x^2 - x - 60 = 0
(x-4)(x+15) =
x = 4 or x = -3.75
OMG thank you so much for the thoroughness
To solve the given equation for x, let's simplify the equation step by step and then apply some algebraic manipulations:
1/x + 1/x + 5 + 1/(3x) = 1/2.25
First, let's find a common denominator for the fractions on the left-hand side. Since we have three terms with x denominators, we can multiply the first term by (3x)/(3x), the second term by (3x)/(3x), and the third term by (x)/(x). Doing this, we get:
(3x) / (3x^2) + (3x) / (3x^2) + 5 + (1 / (3x)) = 1 / 2.25
Now, let's combine the fractions that have the same denominator:
(6x) / (3x^2) + 5 + (1 / (3x)) = 1 / 2.25
Next, let's add the fractions on the left-hand side (6x / 3x^2 and 1 / (3x)) by finding a common denominator. The common denominator is 3x^2:
(6x + 1) / (3x^2) + 5 = 1 / 2.25
Now, let's get rid of the fractions by multiplying both sides of the equation by the common denominator (3x^2):
(3x^2) * ((6x + 1) / (3x^2) + 5) = (3x^2) * (1 / 2.25)
Simplifying the equation gives:
6x + 1 + 15x^2 = (3/2.25) * x^2
To bring all the terms to one side of the equation, subtract (3/2.25) * x^2 from both sides:
15x^2 - (3/2.25) * x^2 + 6x + 1 = 0
Now, let's simplify the equation further:
15x^2 - (3/2.25) * x^2 + 6x + 1 = 0
15x^2 - (1.33) * x^2 + 6x + 1 = 0
15x^2 - 1.33x^2 + 6x + 1 = 0
13.67x^2 + 6x + 1 = 0
Finally, we have a quadratic equation in terms of x which we can solve using the quadratic formula or factoring if possible.