Question

1/4x-3/4=7/x solving the following rational equation for x

Answer 1

Answer:

x¹= -4 , x²= 7

Step-by-step explanation:

1) Determine the defined range

¼x-¾=7/x =/ 0

2) Move the expression to the left-hand side and change its sign

$\frac{1}{4} x - \frac{3}{4} - \frac{7}{x} = 0$

3) Write all numerators above the least common denominator 4x

$\frac{ {x}^{2} - 3x - 28 }{4x} = 0$

4) When the quotient of expressions equals 0, the numerator has to be 0

x²-3x-28=0

5) Write -3x as a difference

x²+4x-7x-28=0

6) Factor out x from the expression

x(x+4)-7x-28=0

7) Factor out -7 from the expression

×(x+4)-7(x+4)=0

8) Factor out x+4 from the expression

(x+4)×(x-7)=0

9) When the product of the factors equals 0, at least one factor is 0

x+4=0

x-7=0

10) Solve for the x

x+4=0 -> x= -4

x-7=0 -> x=7

11) Check if the solution has defined range

x= -4

x=7

12) The solution has 2 solutions

$x1 = - 4 \: \: x2 = 7$