Solve the Complex Fraction Equation: -7/((x+4)×3-7) = 2/(5+x)

To solve the given equation, we'll use the approach of cross-multiplication. Let's work through it step by step:

• Step 1: Simplify the denominators:
• In $(x+4)\times3-7$, first compute the multiplication: $(x+4)\times3 = 3x + 12$.
• Subtract 7, obtaining: $3x + 12 - 7 = 3x + 5$.
• Step 2: Plug these simplifications back into the equation:

$\frac{-7}{3x+5} = \frac{2}{5+x}$

• Step 3: Cross-multiply to clear the fractions:

$-7(5+x) = 2(3x+5)$

• Step 4: Expand both sides:
• $-7 \times 5 = -35$ and $-7 \times x = -7x$, so the left side is $-35 - 7x$.
• For the right side: $2 \times 3x = 6x$ and $2 \times 5 = 10$, so it equals $6x + 10$.
• Step 5: Set up the equation from expansions:

$-35 - 7x = 6x + 10$

• Step 6: Solve for $x$:
• Add $7x$ to both sides to collect $x$ terms on one side:

$-35 = 6x + 10 + 7x$

• This simplifies to: $-35 = 13x + 10$.
• Subtract 10 from both sides:

$-35 - 10 = 13x$

$-45 = 13x$

• Divide both sides by 13 to solve for $x$:

$x = \frac{-45}{13}$

• Convert $-\frac{45}{13}$ to a decimal: $-3.46$ (rounded to two decimal places).
• Step 7: Verify:
• Verify $5 + x \neq 0$ and $3x + 5 \neq 0$ based on our found value, ensuring no division by zero. Both conditions are true.

Therefore, the solution to the problem is $x = -3.46$.