Solve 4/(3x+1) - 5/(7-2x) = 3/4

Question

Solve the equation

Solve for x

x_{1} = \frac{149 - 17089}{36}, x_{2} = \frac{149 + 17089}{36}

Alternative Form

x_{1} \approx 0.507642, x_{2} \approx 7.770136

Evaluate

\frac{4}{3 x + 1} - \frac{5}{7 - 2 x} = \frac{3}{4}

Find the domain

More Steps

\frac{4}{3 x + 1} - \frac{5}{7 - 2 x} = \frac{3}{4}, x \in (- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \frac{7}{2}) \cup (\frac{7}{2}, + \infty)

Multiply both sides of the equation by LCD

(\frac{4}{3 x + 1} - \frac{5}{7 - 2 x}) \times 4 (3 x + 1) (7 - 2 x) = \frac{3}{4} \times 4 (3 x + 1) (7 - 2 x)

Simplify the equation

More Steps

92 - 92 x = \frac{3}{4} \times 4 (3 x + 1) (7 - 2 x)

Simplify the equation

More Steps

92 - 92 x = 57 x - 18 x^{2} + 21

Move the expression to the left side

92 - 92 x - (57 x - 18 x^{2} + 21) = 0

Subtract the terms

More Steps

71 - 149 x + 18 x^{2} = 0

Rewrite in standard form

18 x^{2} - 149 x + 71 = 0

Substitute a= 18,b= - 149 and c= 71 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{149 \pm ( - 149 ) ^{2} - 4 \times 18 \times 71}{2 \times 18}

Simplify the expression

x = \frac{149 \pm ( - 149 ) ^{2} - 4 \times 18 \times 71}{36}

Simplify the expression

More Steps

x = \frac{149 \pm 17089}{36}

Separate the equation into 2 possible cases

x = \frac{149 + 17089}{36} x = \frac{149 - 17089}{36}

Check if the solution is in the defined range

x = \frac{149 + 17089}{36} x = \frac{149 - 17089}{36}, x \in (- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \frac{7}{2}) \cup (\frac{7}{2}, + \infty)

Find the intersection of the solution and the defined range

x = \frac{149 + 17089}{36} x = \frac{149 - 17089}{36}

Solution

x_{1} = \frac{149 - 17089}{36}, x_{2} = \frac{149 + 17089}{36}

Alternative Form

x_{1} \approx 0.507642, x_{2} \approx 7.770136

Show Solution