Solve 3/2x-1-2/1-3x=4

$$\frac { 3 } { 2 x - 1 } - \frac { 2 } { 1 - 3 x } = 4$$

Solve for x

$$x=1$$

$$x=\frac{3}{8}=0.375$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{3},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(3x-1\right), the least common multiple of 2x-1,1-3x.

$$\left(3x-1\right)\times 3-\left(1-2x\right)\times 2=4\left(2x-1\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by 3.

$$9x-3-\left(1-2x\right)\times 2=4\left(2x-1\right)\left(3x-1\right)$$

Use the distributive property to multiply 1-2x by 2.

$$9x-3-\left(2-4x\right)=4\left(2x-1\right)\left(3x-1\right)$$

To find the opposite of 2-4x, find the opposite of each term.

$$9x-3-2+4x=4\left(2x-1\right)\left(3x-1\right)$$

Subtract 2 from -3 to get -5.

$$9x-5+4x=4\left(2x-1\right)\left(3x-1\right)$$

Combine 9x and 4x to get 13x.

$$13x-5=4\left(2x-1\right)\left(3x-1\right)$$

Use the distributive property to multiply 4 by 2x-1.

$$13x-5=\left(8x-4\right)\left(3x-1\right)$$

Use the distributive property to multiply 8x-4 by 3x-1 and combine like terms.

$$13x-5=24x^{2}-20x+4$$

Subtract 24x^{2} from both sides.

$$13x-5-24x^{2}=-20x+4$$

Add 20x to both sides.

$$13x-5-24x^{2}+20x=4$$

Combine 13x and 20x to get 33x.

$$33x-5-24x^{2}=4$$

Subtract 4 from both sides.

$$33x-5-24x^{2}-4=0$$

Subtract 4 from -5 to get -9.

$$33x-9-24x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-24x^{2}+33x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, 33 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-24\right)\left(-9\right)}}{2\left(-24\right)}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\left(-24\right)\left(-9\right)}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-33±\sqrt{1089+96\left(-9\right)}}{2\left(-24\right)}$$

Multiply 96 times -9.

$$x=\frac{-33±\sqrt{1089-864}}{2\left(-24\right)}$$

Add 1089 to -864.

$$x=\frac{-33±\sqrt{225}}{2\left(-24\right)}$$

Take the square root of 225.

$$x=\frac{-33±15}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-33±15}{-48}$$

Now solve the equation x=\frac{-33±15}{-48} when ± is plus. Add -33 to 15.

$$x=-\frac{18}{-48}$$

Reduce the fraction \frac{-18}{-48} to lowest terms by extracting and canceling out 6.

$$x=\frac{3}{8}$$

Now solve the equation x=\frac{-33±15}{-48} when ± is minus. Subtract 15 from -33.

$$x=-\frac{48}{-48}$$

Divide -48 by -48.

$$x=1$$

The equation is now solved.

$$x=\frac{3}{8} x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{3},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(3x-1\right), the least common multiple of 2x-1,1-3x.

$$\left(3x-1\right)\times 3-\left(1-2x\right)\times 2=4\left(2x-1\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by 3.

$$9x-3-\left(1-2x\right)\times 2=4\left(2x-1\right)\left(3x-1\right)$$

Use the distributive property to multiply 1-2x by 2.

$$9x-3-\left(2-4x\right)=4\left(2x-1\right)\left(3x-1\right)$$

To find the opposite of 2-4x, find the opposite of each term.

$$9x-3-2+4x=4\left(2x-1\right)\left(3x-1\right)$$

Subtract 2 from -3 to get -5.

$$9x-5+4x=4\left(2x-1\right)\left(3x-1\right)$$

Combine 9x and 4x to get 13x.

$$13x-5=4\left(2x-1\right)\left(3x-1\right)$$

Use the distributive property to multiply 4 by 2x-1.

$$13x-5=\left(8x-4\right)\left(3x-1\right)$$

Use the distributive property to multiply 8x-4 by 3x-1 and combine like terms.

$$13x-5=24x^{2}-20x+4$$

Subtract 24x^{2} from both sides.

$$13x-5-24x^{2}=-20x+4$$

Add 20x to both sides.

$$13x-5-24x^{2}+20x=4$$

Combine 13x and 20x to get 33x.

$$33x-5-24x^{2}=4$$

Add 5 to both sides.

$$33x-24x^{2}=4+5$$

Add 4 and 5 to get 9.

$$33x-24x^{2}=9$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-24x^{2}+33x=9$$

Divide both sides by -24.

$$\frac{-24x^{2}+33x}{-24}=\frac{9}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$x^{2}+\frac{33}{-24}x=\frac{9}{-24}$$

Reduce the fraction \frac{33}{-24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{11}{8}x=\frac{9}{-24}$$

Reduce the fraction \frac{9}{-24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{11}{8}x=-\frac{3}{8}$$

Divide -\frac{11}{8}, the coefficient of the x term, by 2 to get -\frac{11}{16}. Then add the square of -\frac{11}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{8}x+\left(-\frac{11}{16}\right)^{2}=-\frac{3}{8}+\left(-\frac{11}{16}\right)^{2}$$

Square -\frac{11}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{8}x+\frac{121}{256}=-\frac{3}{8}+\frac{121}{256}$$

Add -\frac{3}{8} to \frac{121}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{8}x+\frac{121}{256}=\frac{25}{256}$$

Factor x^{2}-\frac{11}{8}x+\frac{121}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{16}\right)^{2}=\frac{25}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{16}\right)^{2}}=\sqrt{\frac{25}{256}}$$

Simplify.

$$x-\frac{11}{16}=\frac{5}{16} x-\frac{11}{16}=-\frac{5}{16}$$

Add \frac{11}{16} to both sides of the equation.

$$x=1 x=\frac{3}{8}$$