Solve each rational equation. State the restrictions
𝑥−3 / 3𝑥 −1 / 4 =1 / 2𝑥
To solve the rational equation (𝑥−3)/(3𝑥) −1/4 = 1/(2𝑥), we need to combine the rational expressions on the left side and find a common denominator. The common denominator will be 4𝑥.
Multiplying the first term by (4/4) and the second term by (3/3) gives us:
((4/4)(𝑥−3) − (3/3)(1/4))/(4𝑥) = 1/(2𝑥)
(4𝑥−12 − 3/12)/(4𝑥) = 1/(2𝑥)
Simplifying the numerator gives:
(4𝑥−12 − 3)/(12𝑥) = 1/(2𝑥)
(4𝑥−15)/(12𝑥) = 1/(2𝑥)
Cross multiplying gives:
2𝑥(4𝑥−15) = 12𝑥
Expanding the left side gives:
8𝑥^2 − 30𝑥 = 12𝑥
Rearranging the equation:
8𝑥^2 − 42𝑥 = 0
Factoring out a common factor of 2𝑥:
2𝑥(4𝑥 − 21) = 0
Setting each factor equal to zero and solving for 𝑥, we get two possible solutions:
2𝑥 = 0 or 4𝑥 − 21 = 0
𝑥 = 0 or 4𝑥 = 21
𝑥 = 0 or 𝑥 = 21/4
Therefore, the possible solutions to the equation are 𝑥 = 0 and 𝑥 = 21/4.
Restrictions:
We need to impose a restriction on the equation to ensure we do not divide by zero. In this case, we have two denominators, 3𝑥 and 2𝑥, so we need to make sure these are not zero.
For 3𝑥 ≠ 0, 𝑥 ≠ 0
For 2𝑥 ≠ 0, 𝑥 ≠ 0
Therefore, the restriction is 𝑥 ≠ 0.