Solve each rational equation. State the restrictions.
𝑥−2/2+ 4𝑥−4/5= 2 −3𝑥+6/10
To solve the rational equation 𝑥−2/2+ 4𝑥−4/5 = 2 −3𝑥+6/10, we first need to find a common denominator for all the fractions involved. The least common denominator (LCD) for the fractions 2, 5, and 10 is 10. Rewrite each fraction using the LCD:
(5𝑥−4)/10 + (8𝑥−8)/10 = 20 −3(𝑥−2)/10
Next, combine like terms on each side of the equation:
(5𝑥 + 8𝑥 -4𝑥) / 10 - 2 / 10 = 20 - 30 (𝑥 - 2) / 10
Now simplify the equation:
9𝑥 / 10 - 2 / 10 = 20 - 30𝑥 / 10 + 60 / 10
Now combine like terms again:
(9𝑥 - 2) / 10 = (80 - 30𝑥 + 60) / 10
(9𝑥 - 2) / 10 = (140 - 30𝑥) / 10
Since both sides of the equation are already in simplified form, we can drop the denominators:
9𝑥 - 2 = 140 - 30𝑥
Now isolate the variable by moving all terms involving 𝑥 to one side:
9𝑥 + 30𝑥 = 140 + 2
39𝑥 = 142
Finally, solve for 𝑥 by dividing both sides by 39:
𝑥 = 142 / 39
The solution to the rational equation is 𝑥 = 142 / 39.
However, there may be restrictions on this solution. To find the restrictions, we need to identify any values of 𝑥 that would make any denominators equal to zero. In this equation, the only denominator present is 10. So we must set the denominator equal to zero and solve for 𝑥:
10 = 0
Since the equation 10 = 0 has no solutions, there are no restrictions on the domain; the solution 𝑥 = 142 / 39 is valid for all values of 𝑥.