1. Solve each inequality and express the solution in interval notation. (a) Solve $$-4[x + 2(3 - x)] \leq 5(2 - 4x) + 6$$ Step 1: Expand terms inside the brackets. $$-4[x + 6 - 2x] \leq 10 - 20x + 6$$ Step 2: Simplify inside the bracket. $$-4[6 - x] \leq 16 - 20x$$ Step 3: Distribute \(-4\). $$-24 + 4x \leq 16 - 20x$$ Step 4: Bring like terms together. $$4x + 20x \leq 16 + 24$$ $$24x \leq 40$$ Step 5: Divide both sides by 24. $$x \leq \frac{40}{24} = \frac{5}{3}$$ Solution in interval notation: $$(-\infty, \frac{5}{3}]$$ --- (b) Solve $$-\frac{3}{4}(x - 2) + \frac{2}{3}(x + 5) > \frac{1}{6}x$$ Step 1: Distribute each term. $$-\frac{3}{4}x + \frac{3}{2} + \frac{2}{3}x + \frac{10}{3} > \frac{1}{6}x$$ Step 2: Combine like terms on the left. $$\left(-\frac{3}{4}x + \frac{2}{3}x\right) + \left(\frac{3}{2} + \frac{10}{3}\right) > \frac{1}{6}x$$ Step 3: Find common denominators for the coefficients. $$-\frac{9}{12}x + \frac{8}{12}x = -\frac{1}{12}x$$ $$\frac{3}{2} = \frac{9}{6}, \frac{10}{3} = \frac{20}{6}, \frac{9}{6} + \frac{20}{6} = \frac{29}{6}$$ Step 4: Rewrite inequality. $$-\frac{1}{12}x + \frac{29}{6} > \frac{1}{6}x$$ Step 5: Bring all terms involving \(x\) to one side. $$-\frac{1}{12}x - \frac{1}{6}x > -\frac{29}{6}$$ Step 6: Find common denominator and combine. $$-\frac{1}{12}x - \frac{2}{12}x = -\frac{3}{12}x = -\frac{1}{4}x$$ So: $$-\frac{1}{4}x > -\frac{29}{6}$$ Step 7: Multiply both sides by \(-4\), reverse inequality because of negative: $$x < \frac{29}{6} \times 4 = \frac{116}{6} = \frac{58}{3}$$ Solution in interval notation: $$(-\infty, \frac{58}{3})$$ --- (c) Solve the compound inequality: $$-\frac{3}{4} \leq \frac{5 - 2x}{6} < \frac{2}{3}$$ Step 1: Split into two inequalities: $$-\frac{3}{4} \leq \frac{5 - 2x}{6} \quad \text{and} \quad \frac{5 - 2x}{6} < \frac{2}{3}$$ First inequality: Multiply both sides by 6: $$6 \times -\frac{3}{4} \leq 5 - 2x$$ $$-\frac{18}{4} \leq 5 - 2x$$ $$-\frac{9}{2} \leq 5 - 2x$$ Bring 5 to left: $$-\frac{9}{2} - 5 \leq -2x$$ $$-\frac{9}{2} - \frac{10}{2} = -\frac{19}{2} \leq -2x$$ Divide by \(-2\) and reverse inequality: $$x \leq \frac{19}{4}$$ Second inequality: Multiply both sides by 6: $$5 - 2x < 4$$ Bring 5 to right: $$-2x < -1$$ Divide by \(-2\) and reverse inequality: $$x > \frac{1}{2}$$ Step 2: Combine results. $$\frac{1}{2} < x \leq \frac{19}{4}$$ Solution in interval notation: $$\left(\frac{1}{2}, \frac{19}{4}\right]$$ --- 2. Find required production for fourth month to meet average target. Step 1: Let the fourth month production be $$x$$. Step 2: Average over 4 months must be at least 1200 units. $$\frac{1150 + 1180 + 1210 + x}{4} \geq 1200$$ Step 3: Sum known productions. $$\frac{3540 + x}{4} \geq 1200$$ Step 4: Multiply both sides by 4. $$3540 + x \geq 4800$$ Step 5: Solve for $$x$$. $$x \geq 4800 - 3540 = 1260$$ Final answer: The company must produce at least 1260 units in the 4th month.