1. **State the problem:** Solve the equation $$\frac{x + 1}{5} = \frac{7 - x}{6x}$$ for $x$. 2. **Clear the denominators:** Multiply both sides by $30x$ (the least common multiple of $5$ and $6x$) to eliminate fractions: $$30x \times \frac{x + 1}{5} = 30x \times \frac{7 - x}{6x}$$ 3. **Simplify both sides:** Left side: $$30x \times \frac{x + 1}{5} = 6x(x + 1) = 6x^2 + 6x$$ Right side: $$30x \times \frac{7 - x}{6x} = 5(7 - x) = 35 - 5x$$ 4. **Set up the equation:** $$6x^2 + 6x = 35 - 5x$$ 5. **Bring all terms to one side:** $$6x^2 + 6x + 5x - 35 = 0$$ Simplify: $$6x^2 + 11x - 35 = 0$$ 6. **Factor the quadratic:** Find two numbers that multiply to $6 \times (-35) = -210$ and add to $11$. These numbers are $21$ and $-10$ because $21 \times (-10) = -210$ and $21 + (-10) = 11$. Rewrite the middle term: $$6x^2 + 21x - 10x - 35 = 0$$ Group terms: $$(6x^2 + 21x) + (-10x - 35) = 0$$ Factor each group: $$3x(2x + 7) - 5(2x + 7) = 0$$ Factor out the common binomial: $$(3x - 5)(2x + 7) = 0$$ 7. **Solve each factor for $x$:** $$3x - 5 = 0 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3}$$ $$2x + 7 = 0 \Rightarrow 2x = -7 \Rightarrow x = -\frac{7}{2}$$ **Final answer:** $$x = \frac{5}{3} \quad \text{or} \quad x = -\frac{7}{2}$$