1. **State the problem:** Solve the inequality and equation system: $$3(2+x) \leq 4x = 1$$. 2. **Analyze the problem:** The expression seems to combine an inequality and an equation. We interpret it as two separate conditions: - Inequality: $$3(2+x) \leq 4x$$ - Equation: $$4x = 1$$ 3. **Solve the equation first:** $$4x = 1$$ Divide both sides by 4: $$x = \frac{1}{4}$$ 4. **Check if this value satisfies the inequality:** Substitute $$x = \frac{1}{4}$$ into $$3(2+x) \leq 4x$$: $$3\left(2 + \frac{1}{4}\right) \leq 4 \times \frac{1}{4}$$ Simplify inside the parentheses: $$3 \times \frac{9}{4} \leq 1$$ Multiply: $$\frac{27}{4} \leq 1$$ 5. **Evaluate the inequality:** $$\frac{27}{4} = 6.75$$ which is not less than or equal to 1. 6. **Conclusion:** The value $$x=\frac{1}{4}$$ satisfies the equation but does not satisfy the inequality. 7. **Solve the inequality separately:** $$3(2+x) \leq 4x$$ Expand: $$6 + 3x \leq 4x$$ Subtract $$3x$$ from both sides: $$6 + \cancel{3x} \leq 4x - \cancel{3x}$$ $$6 \leq x$$ 8. **Final solution:** - Equation solution: $$x = \frac{1}{4}$$ - Inequality solution: $$x \geq 6$$ Since the equation and inequality cannot be true simultaneously, there is no $$x$$ that satisfies both at the same time.