Solve 1.54x/3(x+4)-1/2(x-1)-1/2=0

$$1.5 \frac { 4 x } { 3 ( x + 4 ) } - \frac { 1 } { 2 ( x - 1 ) } - \frac { 1 } { 2 } = 0$$

Solve for x

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+4\right), the least common multiple of 3\left(x+4\right),2\left(x-1\right),2.

$$1.5\left(2x-2\right)\times 4x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Multiply 1.5 and 4 to get 6.

$$6\left(2x-2\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Use the distributive property to multiply 6 by 2x-2.

$$\left(12x-12\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Use the distributive property to multiply 12x-12 by x.

$$12x^{2}-12x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

To find the opposite of 3x+12, find the opposite of each term.

$$12x^{2}-12x-3x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Combine -12x and -3x to get -15x.

$$12x^{2}-15x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Multiply 6 and -\frac{1}{2} to get -3.

$$12x^{2}-15x-12-3\left(x-1\right)\left(x+4\right)=0$$

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

$$12x^{2}-15x-12+\left(-3x+3\right)\left(x+4\right)=0$$

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

$$12x^{2}-15x-12-3x^{2}-9x+12=0$$

Combine 12x^{2} and -3x^{2} to get 9x^{2}.

$$9x^{2}-15x-12-9x+12=0$$

Combine -15x and -9x to get -24x.

$$9x^{2}-24x-12+12=0$$

Add -12 and 12 to get 0.

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

Factor out x.

$$x\left(9x-24\right)=0$$

To find equation solutions, solve x=0 and 9x-24=0.

$$x=0 x=\frac{8}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+4\right), the least common multiple of 3\left(x+4\right),2\left(x-1\right),2.

$$1.5\left(2x-2\right)\times 4x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Multiply 1.5 and 4 to get 6.

$$6\left(2x-2\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Use the distributive property to multiply 6 by 2x-2.

$$\left(12x-12\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Use the distributive property to multiply 12x-12 by x.

$$12x^{2}-12x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

To find the opposite of 3x+12, find the opposite of each term.

$$12x^{2}-12x-3x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Combine -12x and -3x to get -15x.

$$12x^{2}-15x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Multiply 6 and -\frac{1}{2} to get -3.

$$12x^{2}-15x-12-3\left(x-1\right)\left(x+4\right)=0$$

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

$$12x^{2}-15x-12+\left(-3x+3\right)\left(x+4\right)=0$$

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

$$12x^{2}-15x-12-3x^{2}-9x+12=0$$

Combine 12x^{2} and -3x^{2} to get 9x^{2}.

$$9x^{2}-15x-12-9x+12=0$$

Combine -15x and -9x to get -24x.

$$9x^{2}-24x-12+12=0$$

Add -12 and 12 to get 0.

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

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

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

Take the square root of \left(-24\right)^{2}.

$$x=\frac{-\left(-24\right)±24}{2\times 9}$$

The opposite of -24 is 24.

$$x=\frac{24±24}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{24±24}{18}$$

Now solve the equation x=\frac{24±24}{18} when ± is plus. Add 24 to 24.

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

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

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

Now solve the equation x=\frac{24±24}{18} when ± is minus. Subtract 24 from 24.

$$x=\frac{0}{18}$$

Divide 0 by 18.

$$x=0$$

The equation is now solved.

$$x=\frac{8}{3} x=0$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+4\right), the least common multiple of 3\left(x+4\right),2\left(x-1\right),2.

$$1.5\left(2x-2\right)\times 4x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Multiply 1.5 and 4 to get 6.

$$6\left(2x-2\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Use the distributive property to multiply 6 by 2x-2.

$$\left(12x-12\right)x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Use the distributive property to multiply 12x-12 by x.

$$12x^{2}-12x-\left(3x+12\right)+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

To find the opposite of 3x+12, find the opposite of each term.

$$12x^{2}-12x-3x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Combine -12x and -3x to get -15x.

$$12x^{2}-15x-12+6\left(x-1\right)\left(x+4\right)\left(-\frac{1}{2}\right)=0$$

Multiply 6 and -\frac{1}{2} to get -3.

$$12x^{2}-15x-12-3\left(x-1\right)\left(x+4\right)=0$$

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

$$12x^{2}-15x-12+\left(-3x+3\right)\left(x+4\right)=0$$

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

$$12x^{2}-15x-12-3x^{2}-9x+12=0$$

Combine 12x^{2} and -3x^{2} to get 9x^{2}.

$$9x^{2}-15x-12-9x+12=0$$

Combine -15x and -9x to get -24x.

$$9x^{2}-24x-12+12=0$$

Add -12 and 12 to get 0.

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

Divide both sides by 9.

$$\frac{9x^{2}-24x}{9}=\frac{0}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{24}{9}\right)x=\frac{0}{9}$$

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

$$x^{2}-\frac{8}{3}x=\frac{0}{9}$$

Divide 0 by 9.

$$x^{2}-\frac{8}{3}x=0$$

Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=\left(-\frac{4}{3}\right)^{2}$$

Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{16}{9}$$

Factor x^{2}-\frac{8}{3}x+\frac{16}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

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

Add \frac{4}{3} to both sides of the equation.

$$x=\frac{8}{3} x=0$$