Questions: Solve the following equation for (x) : [ frac2x=-frac7x-4 ] Answer as a proper or improper fraction simplified in lowest terms. Provide your answer below: [ x= ]

$Solve the following equation for (x) : [ frac2x=-frac7x-4 ] Answer as a proper or improper fraction simplified in lowest terms. Provide your answer below: [ x= ]$

Transcript text: Solve the following equation for $x$ : \[ \frac{2}{x}=-\frac{7}{x-4} \] Answer as a proper or improper fraction simplified in lowest terms. Provide your answer below: \[ x=\square \]

Solution

Solution Steps

Step 1: Cross-Multiply

To solve the equation

\[ \frac{2}{x} = -\frac{7}{x-4} \]

we start by cross-multiplying to eliminate the fractions:

\[ 2(x - 4) = -7x \]

Step 2: Distribute and Simplify

Distribute the 2 on the left side of the equation:

\[ 2x - 8 = -7x \]

Step 3: Combine Like Terms

Add \(7x\) to both sides to combine like terms:

\[ 2x + 7x - 8 = 0 \]

\[ 9x - 8 = 0 \]

Step 4: Solve for \(x\)

Add 8 to both sides:

\[ 9x = 8 \]

Divide both sides by 9 to solve for \(x\):

\[ x = \frac{8}{9} \]

Final Answer

The solution to the equation is

\[ \boxed{x = \frac{8}{9}} \]

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