Solve each rational equation. State the restrictions 𝑥−2/2 +4𝑥−4/5= 2 − 3𝑥+6/10
To solve the rational equation, we first need to simplify both sides of the equation.
Starting with the left side:
(𝑥−2)/2 + (4𝑥−4)/5
To add these fractions, we need a common denominator. The least common multiple of 2 and 5 is 10.
So, multiplying the first fraction by 5/5 and the second fraction by 2/2:
(5(𝑥−2))/10 + (2(4𝑥−4))/10
Simplifying this gives:
(5𝑥−10)/10 + (8𝑥−8)/10
Now, we can combine the two fractions:
(5𝑥+8𝑥−10−8)/10
(13𝑥−18)/10
Simplifying the right side of the equation:
2 − (3𝑥+6)/10
Multiplying the numerator and denominator of the second fraction by -1 to change the sign of both terms gives:
2 − (-1(3𝑥+6))/10
2 + (3𝑥+6)/10
Now, we have the equation in simplified form:
(13𝑥−18)/10 = 2 + (3𝑥+6)/10
To get rid of the denominators, we can multiply both sides of the equation by 10:
10((13𝑥−18)/10) = 10(2 + (3𝑥+6)/10)
Simplifying gives:
13𝑥−18 = 20 + 3𝑥+6
Next, we can combine like terms on the right side of the equation:
13𝑥−18 = 26 + 3𝑥
Moving all the 𝑥 terms to one side and the constant terms to the other side:
13𝑥−3𝑥 = 26 + 18
10𝑥 = 44
Finally, we can solve for 𝑥 by dividing both sides of the equation by 10:
𝑥 = 44/10
𝑥 = 4.4
The solution to the rational equation is 𝑥 = 4.4. However, we need to check for any restrictions on 𝑥.
The only potential restriction in this equation is if the denominator of any fraction becomes zero. Looking back at the original equation: (𝑥−2)/2 + (4𝑥−4)/5 = 2 − 3𝑥+6/10
The denominators are 2 and 5. Neither of these is zero, so there are no restrictions on 𝑥.
Therefore, the solution to the rational equation is 𝑥 = 4.4, with no restrictions.