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Question Solve the equation 8x+76x=−56\frac{8}{x}+\frac{7}{6 x}=\frac{-5}{6} for xx.

Studdy Solution

STEP 1

1. The equation 8x+76x=−56\frac{8}{x}+\frac{7}{6x}=\frac{-5}{6} involves rational expressions, and xx is not equal to zero.
2. The equation can be solved by finding a common denominator and combining terms.
3. The solution will be a value or values of xx that satisfy the given equation.

STEP 2

1. Find a common denominator for the rational expressions.
2. Combine the terms over the common denominator.
3. Solve the resulting equation for xx.

STEP 3

Identify the least common denominator (LCD) for the fractions 8x\frac{8}{x} and 76x\frac{7}{6x}.
The LCD for xx and 6x6x is 6x6x.

STEP 4

Rewrite each fraction with the common denominator 6x6x.
8x=8⋅66x \frac{8}{x} = \frac{8 \cdot 6}{6x} 76x=76x \frac{7}{6x} = \frac{7}{6x}

STEP 5

Combine the fractions over the common denominator.
8⋅66x+76x=486x+76x \frac{8 \cdot 6}{6x} + \frac{7}{6x} = \frac{48}{6x} + \frac{7}{6x}

STEP 6

Add the numerators and keep the common denominator.
486x+76x=48+76x \frac{48}{6x} + \frac{7}{6x} = \frac{48 + 7}{6x}

STEP 7

Simplify the numerator.
48+76x=556x \frac{48 + 7}{6x} = \frac{55}{6x}

STEP 8

Set the combined fraction equal to the right-hand side of the original equation and solve for xx.
556x=−56 \frac{55}{6x} = \frac{-5}{6}

STEP 9

Cross-multiply to eliminate the fractions.
55⋅6=−5⋅6x 55 \cdot 6 = -5 \cdot 6x

STEP 10

Simplify the equation.
330=−30x 330 = -30x

STEP 11

Divide both sides by −30-30 to solve for xx.
x=330−30 x = \frac{330}{-30}

STEP 12

Simplify the fraction to find the value of xx.
x=−11 x = -11
The solution to the equation 8x+76x=−56\frac{8}{x}+\frac{7}{6x}=\frac{-5}{6} is x=−11x = -11.

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