1. **Stating the problem:** We are given the efficiency function of two weaving machines combined: $$E(x) = \frac{3x + 6}{x - 1}$$ We need to find the values of $x$ such that the efficiency does not exceed 9, i.e., $$E(x) \leq 9$$ 2. **Formula and rules:** We will solve the inequality: $$\frac{3x + 6}{x - 1} \leq 9$$ Important rules: - The denominator $x - 1 \neq 0$ (so $x \neq 1$). - When multiplying both sides by an expression involving $x$, consider the sign of that expression to avoid reversing the inequality incorrectly. 3. **Solving the inequality:** Start with: $$\frac{3x + 6}{x - 1} \leq 9$$ Rewrite as: $$\frac{3x + 6}{x - 1} - 9 \leq 0$$ Find common denominator: $$\frac{3x + 6 - 9(x - 1)}{x - 1} \leq 0$$ Simplify numerator: $$3x + 6 - 9x + 9 = -6x + 15$$ So inequality becomes: $$\frac{-6x + 15}{x - 1} \leq 0$$ Or equivalently: $$\frac{15 - 6x}{x - 1} \leq 0$$ 4. **Find critical points:** - Numerator zero when $15 - 6x = 0 \Rightarrow x = \frac{15}{6} = 2.5$ - Denominator zero when $x - 1 = 0 \Rightarrow x = 1$ 5. **Test intervals determined by critical points:** Intervals: $(-\infty, 1)$, $(1, 2.5)$, $(2.5, \infty)$ - For $x < 1$ (e.g., $x=0$): numerator $15 - 0 = 15 > 0$, denominator $0 - 1 = -1 < 0$, fraction $>0 / <0 = <$ 0? No, positive over negative is negative, so fraction is negative, which satisfies $\leq 0$. - For $1 < x < 2.5$ (e.g., $x=2$): numerator $15 - 12 = 3 > 0$, denominator $2 - 1 = 1 > 0$, fraction positive over positive = positive, which does not satisfy $\leq 0$. - For $x > 2.5$ (e.g., $x=3$): numerator $15 - 18 = -3 < 0$, denominator $3 - 1 = 2 > 0$, fraction negative over positive = negative, satisfies $\leq 0$. 6. **Check points where numerator or denominator is zero:** - At $x=2.5$, numerator zero, fraction zero, satisfies $\leq 0$. - At $x=1$, denominator zero, undefined, exclude. 7. **Final solution:** $$(-\infty, 1) \cup [2.5, \infty)$$ **Answer:** The values of $x$ for which $E(x) \leq 9$ are $$x \in (-\infty, 1) \cup [2.5, \infty)$$