1) Solve $\frac{x - 7}{x - 1} < 0$. Step 1: Identify critical points where numerator or denominator is zero: $x=7$ and $x=1$. Step 2: These points divide the number line into intervals: $(-\infty,1)$, $(1,7)$, $(7,\infty)$. Step 3: Test each interval: - For $x=0$ in $(-\infty,1)$: $\frac{0-7}{0-1} = \frac{-7}{-1} = 7 > 0$ (False) - For $x=2$ in $(1,7)$: $\frac{2-7}{2-1} = \frac{-5}{1} = -5 < 0$ (True) - For $x=8$ in $(7,\infty)$: $\frac{8-7}{8-1} = \frac{1}{7} > 0$ (False) Step 4: Exclude $x=1$ (denominator zero), solution is $(1,7)$. 2) Solve $\frac{x + 5}{x - 4} \leq 0$. Critical points: $x=-5$, $x=4$. Intervals: $(-\infty,-5)$, $(-5,4)$, $(4,\infty)$. Test: - $x=-6$: $\frac{-6+5}{-6-4} = \frac{-1}{-10} = 0.1 > 0$ (False) - $x=0$: $\frac{0+5}{0-4} = \frac{5}{-4} = -1.25 < 0$ (True) - $x=5$: $\frac{5+5}{5-4} = \frac{10}{1} = 10 > 0$ (False) Include $x=-5$ (numerator zero), exclude $x=4$ (denominator zero). Solution: $[-5,4)$. 3) Solve $\frac{x + 32}{x + 6} \leq 3$. Rewrite: $\frac{x + 32}{x + 6} - 3 \leq 0$. Common denominator: $\frac{x + 32 - 3(x + 6)}{x + 6} \leq 0$. Simplify numerator: $x + 32 - 3x - 18 = -2x + 14$. Inequality: $\frac{-2x + 14}{x + 6} \leq 0$. Critical points: numerator zero at $x=7$, denominator zero at $x=-6$. Intervals: $(-\infty,-6)$, $(-6,7)$, $(7,\infty)$. Test: - $x=-7$: numerator $-2(-7)+14=14+14=28>0$, denominator $-7+6=-1<0$, fraction negative (True) - $x=0$: numerator $-2(0)+14=14>0$, denominator $0+6=6>0$, fraction positive (False) - $x=8$: numerator $-2(8)+14=-16+14=-2<0$, denominator $8+6=14>0$, fraction negative (True) Include $x=7$ (numerator zero), exclude $x=-6$. Solution: $(-\infty,-6) \cup [7,\infty)$. 4) Solve $\frac{x + 68}{x + 8} \geq 5$. Rewrite: $\frac{x + 68}{x + 8} - 5 \geq 0$. Common denominator: $\frac{x + 68 - 5(x + 8)}{x + 8} \geq 0$. Simplify numerator: $x + 68 - 5x - 40 = -4x + 28$. Critical points: numerator zero at $x=7$, denominator zero at $x=-8$. Intervals: $(-\infty,-8)$, $(-8,7)$, $(7,\infty)$. Test: - $x=-9$: numerator $-4(-9)+28=36+28=64>0$, denominator $-9+8=-1<0$, fraction negative (False) - $x=0$: numerator $-4(0)+28=28>0$, denominator $0+8=8>0$, fraction positive (True) - $x=8$: numerator $-4(8)+28=-32+28=-4<0$, denominator $8+8=16>0$, fraction negative (False) Include $x=7$, exclude $x=-8$. Solution: $(-8,7]$. 5) Solve $\frac{(x + 3)(x + 5)}{x + 2} \geq 0$. Critical points: zeros of numerator at $x=-3$, $x=-5$, zero of denominator at $x=-2$. Intervals: $(-\infty,-5)$, $(-5,-3)$, $(-3,-2)$, $(-2,\infty)$. Test: - $x=-6$: numerator positive (since both factors negative, product positive), denominator negative, fraction negative (False) - $x=-4$: numerator negative (one factor positive, one negative), denominator negative, fraction positive (True) - $x=-2.5$: numerator negative, denominator negative, fraction positive (True) - $x=0$: numerator positive, denominator positive, fraction positive (True) Include zeros of numerator ($x=-5,-3$), exclude $x=-2$. Solution: $[-5,-3] \cup (-3,-2) \cup (-2,\infty)$. 6) Solve $\frac{x + 6}{x^2 - 5x - 24} \geq 0$. Factor denominator: $x^2 - 5x - 24 = (x - 8)(x + 3)$. Critical points: numerator zero at $x=-6$, denominator zeros at $x=8$, $x=-3$. Intervals: $(-\infty,-6)$, $(-6,-3)$, $(-3,8)$, $(8,\infty)$. Test: - $x=-7$: numerator negative, denominator positive, fraction negative (False) - $x=-5$: numerator positive, denominator positive, fraction positive (True) - $x=0$: numerator positive, denominator negative, fraction negative (False) - $x=9$: numerator positive, denominator positive, fraction positive (True) Include $x=-6$, exclude $x=-3,8$. Solution: $[-6,-3) \cup (8,\infty)$. 7) Solve $-\frac{10}{x - 5} \geq -\frac{11}{x - 6}$. Rewrite: $-\frac{10}{x - 5} + \frac{11}{x - 6} \geq 0$. Common denominator: $(x - 5)(x - 6)$. Numerator: $-10(x - 6) + 11(x - 5) = -10x + 60 + 11x - 55 = x + 5$. Inequality: $\frac{x + 5}{(x - 5)(x - 6)} \geq 0$. Critical points: $x=-5$, $x=5$, $x=6$. Intervals: $(-\infty,-5)$, $(-5,5)$, $(5,6)$, $(6,\infty)$. Test: - $x=-6$: numerator negative, denominator positive, fraction negative (False) - $x=0$: numerator positive, denominator positive, fraction positive (True) - $x=5.5$: numerator positive, denominator negative, fraction negative (False) - $x=7$: numerator positive, denominator positive, fraction positive (True) Include $x=-5$, exclude $x=5,6$. Solution: $[-5,5) \cup (6,\infty)$. 8) Solve $-\frac{3}{x + 7} \leq -\frac{4}{x + 8}$. Rewrite: $-\frac{3}{x + 7} + \frac{4}{x + 8} \leq 0$. Common denominator: $(x + 7)(x + 8)$. Numerator: $-3(x + 8) + 4(x + 7) = -3x - 24 + 4x + 28 = x + 4$. Inequality: $\frac{x + 4}{(x + 7)(x + 8)} \leq 0$. Critical points: $x=-4$, $x=-7$, $x=-8$. Intervals: $(-\infty,-8)$, $(-8,-7)$, $(-7,-4)$, $(-4,\infty)$. Test: - $x=-9$: numerator negative, denominator positive, fraction negative (True) - $x=-7.5$: numerator negative, denominator negative, fraction positive (False) - $x=-5$: numerator negative, denominator negative, fraction positive (False) - $x=0$: numerator positive, denominator positive, fraction positive (False) Include $x=-4$, exclude $x=-7,-8$. Solution: $(-\infty,-8) \cup [-4,-7)$. 9) Solve $-\frac{7}{x + 5} \leq -\frac{8}{x + 6}$. Rewrite: $-\frac{7}{x + 5} + \frac{8}{x + 6} \leq 0$. Common denominator: $(x + 5)(x + 6)$. Numerator: $-7(x + 6) + 8(x + 5) = -7x - 42 + 8x + 40 = x - 2$. Inequality: $\frac{x - 2}{(x + 5)(x + 6)} \leq 0$. Critical points: $x=2$, $x=-5$, $x=-6$. Intervals: $(-\infty,-6)$, $(-6,-5)$, $(-5,2)$, $(2,\infty)$. Test: - $x=-7$: numerator negative, denominator positive, fraction negative (True) - $x=-5.5$: numerator negative, denominator negative, fraction positive (False) - $x=0$: numerator negative, denominator negative, fraction positive (False) - $x=3$: numerator positive, denominator positive, fraction positive (False) Include $x=2$, exclude $x=-5,-6$. Solution: $(-\infty,-6) \cup [2,-5)$. 10) Solve $\frac{(x + 7)(x - 3)}{(x - 5)^2} > 0$. Critical points: numerator zeros at $x=-7$, $x=3$, denominator zero at $x=5$. Note denominator squared always positive except at $x=5$ where undefined. Intervals: $(-\infty,-7)$, $(-7,3)$, $(3,5)$, $(5,\infty)$. Test: - $x=-8$: numerator positive, denominator positive, fraction positive (True) - $x=0$: numerator negative, denominator positive, fraction negative (False) - $x=4$: numerator negative, denominator positive, fraction negative (False) - $x=6$: numerator positive, denominator positive, fraction positive (True) Exclude $x=5$. Solution: $(-\infty,-7) \cup (5,\infty)$. 11) Write a rational inequality with solution $(-2,-1) \cup (1,\infty)$. Choose zeros at $x=-2$, $x=-1$, $x=1$. Example: $\frac{(x+2)(x-1)}{x+1} > 0$. Check intervals: - $(-\infty,-2)$: numerator positive, denominator negative, fraction negative (False) - $(-2,-1)$: numerator negative, denominator negative, fraction positive (True) - $(-1,1)$: numerator negative, denominator positive, fraction negative (False) - $(1,\infty)$: numerator positive, denominator positive, fraction positive (True) Solution matches. Final answers: 1) $(1,7)$ 2) $[-5,4)$ 3) $(-\infty,-6) \cup [7,\infty)$ 4) $(-8,7]$ 5) $[-5,-3] \cup (-3,-2) \cup (-2,\infty)$ 6) $[-6,-3) \cup (8,\infty)$ 7) $[-5,5) \cup (6,\infty)$ 8) $(-\infty,-8) \cup [-4,-7)$ 9) $(-\infty,-6) \cup [2,-5)$ 10) $(-\infty,-7) \cup (5,\infty)$ 11) $\frac{(x+2)(x-1)}{x+1} > 0$