Solve (6)/(2x+3)+(4)/(x-2)=(7)/(x+6) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ 6 }{ 2x+3 } + \frac{ 4 }{ x-2 } = \frac{ 7 }{ x+6 }$$

Solve for x

$x=-\frac{6}{13}\approx -0.461538462$

Steps for Solving Linear Equation

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

$$\left(-2+x\right)\left(x+6\right)\times 6+\left(2x+3\right)\left(x+6\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply $-2+x$ by $x+6$ and combine like terms.

$$\left(4x-12+x^{2}\right)\times 6+\left(2x+3\right)\left(x+6\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply $4x-12+x^{2}$ by $6$.

$$24x-72+6x^{2}+\left(2x+3\right)\left(x+6\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply $2x+3$ by $x+6$ and combine like terms.

$$24x-72+6x^{2}+\left(2x^{2}+15x+18\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply $2x^{2}+15x+18$ by $4$.

$$24x-72+6x^{2}+8x^{2}+60x+72=\left(-2+x\right)\left(2x+3\right)\times 7$$

Combine $6x^{2}$ and $8x^{2}$ to get $14x^{2}$.

$$24x-72+14x^{2}+60x+72=\left(-2+x\right)\left(2x+3\right)\times 7$$

Combine $24x$ and $60x$ to get $84x$.

$$84x-72+14x^{2}+72=\left(-2+x\right)\left(2x+3\right)\times 7$$

Add $-72$ and $72$ to get $0$.

$$84x+14x^{2}=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply $-2+x$ by $2x+3$ and combine like terms.

$$84x+14x^{2}=\left(-x-6+2x^{2}\right)\times 7$$

Use the distributive property to multiply $-x-6+2x^{2}$ by $7$.

$$84x+14x^{2}=-7x-42+14x^{2}$$

Add $7x$ to both sides.

$$84x+14x^{2}+7x=-42+14x^{2}$$

Combine $84x$ and $7x$ to get $91x$.

$$91x+14x^{2}=-42+14x^{2}$$

Subtract $14x^{2}$ from both sides.

$$91x+14x^{2}-14x^{2}=-42$$

Combine $14x^{2}$ and $-14x^{2}$ to get $0$.

$$91x=-42$$

Divide both sides by $91$.

$$x=\frac{-42}{91}$$

Reduce the fraction $\frac{-42}{91}$ to lowest terms by extracting and canceling out $7$.

$$x=-\frac{6}{13}$$