Solve each rational equation. State the restrictions
x−2/2 + 4𝑥−4/5 = 2 − 3𝑥+6/10
To solve the rational equation, we will first find a common denominator for all the fractions in the equation. The denominators are 2, 5, and 10. The least common multiple of these three numbers is 10.
Now, we will rewrite the equation with the common denominator of 10:
(5(x - 2) + 4(4x - 4))/10 = (2 - 3(x + 6))/10
Simplifying the numerators, we get:
(5x - 10 + 16x - 16)/10 = (2 - 3x - 18)/10
Combining like terms, we have:
(21x - 26)/10 = (-3x - 16)/10
To clear the fractions, we can multiply both sides of the equation by 10:
10(21x - 26)/10 = 10(-3x - 16)/10
21x - 26 = -3x - 16
Next, we will collect the x-terms on one side of the equation and the constant terms on the other side:
21x + 3x = -16 + 26
24x = 10
Dividing both sides by 24, we get:
x = 10/24
Simplifying the fraction, we have:
x = 5/12
Therefore, the solution to the rational equation is x = 5/12.
Restrictions:
There are no restrictions on the variable x in this equation.