Solve the following equations:

$\displaystyle \text{Question 1: } 4x-9=2x+7$

$\displaystyle 4x-9=2x+7$

$\displaystyle \Rightarrow 4x-2x=7+9$

$\displaystyle \Rightarrow 2x=16$

$\displaystyle \Rightarrow x=8$

$\displaystyle \\$

$\displaystyle \text{Question 2: } 5x+18=11-2x$

$\displaystyle 5x+18=11-2x$

$\displaystyle \Rightarrow 5x+2x=11-18$

$\displaystyle \Rightarrow 7x=-7$

$\displaystyle \Rightarrow x=-1$

$\displaystyle \\$

$\displaystyle \text{Question 3: } 21-3(x-7)=x+20$

$\displaystyle 21-3(x-7)=x+20$

$\displaystyle \Rightarrow 21-3x+21=x+20$

$\displaystyle \Rightarrow 21+21-20=4x$

$\displaystyle \Rightarrow 22=4x$

$\displaystyle \Rightarrow x=5.4$

$\displaystyle \\$

$\displaystyle \text{Question 4: } 3(x-7)-2(3x-4)=(2-5x)$

$\displaystyle 3(x-7)-2(3x-4)=(2-5x)$

$\displaystyle \Rightarrow 3x-21-6x+8=2-5x$

$\displaystyle \Rightarrow -3x-13=2-5x$

$\displaystyle \Rightarrow 2x=15$

$\displaystyle \Rightarrow x=7.5$

$\displaystyle \\$

$\displaystyle \text{Question 5: } 3(x-5)-16x=12-2(x-3)$

$\displaystyle 3(x-5)-16x=12-2(x-3)$

$\displaystyle \Rightarrow 3x-15-16x=12-2x+6$

$\displaystyle \Rightarrow -13x-15=18-2x$

$\displaystyle \Rightarrow 11x=-33$

$\displaystyle \Rightarrow x= -3$

$\displaystyle \\$

$\displaystyle \text{Question 6: } \frac{3x}{4} - \frac{(x-4)}{3} = \frac{5}{3}$

$\displaystyle \frac{3x}{4} - \frac{(x-4)}{3} = \frac{5}{3}$

$\displaystyle \Rightarrow \frac{9x-4(x-4)}{12} = \frac{5}{3}$

$\displaystyle \Rightarrow 27x-12(x-4)=60$

$\displaystyle \Rightarrow 27x-12x+48=60$

$\displaystyle \Rightarrow 15x=12$

$\displaystyle \Rightarrow x= \frac{4}{5}$

$\displaystyle \\$

$\displaystyle \text{Question 7: } \frac{(4x+1)}{3} + \frac{(2x-1)}{2} - \frac{(3x-7)}{5} =6$

$\displaystyle \frac{(4x+1)}{3} + \frac{(2x-1)}{2} - \frac{(3x-7)}{5} =6$

$\displaystyle \Rightarrow 10(4x+1)+15(2x-1)-6(3x-7)=180$

$\displaystyle \Rightarrow 40x+10+30x-15-18x+42=180$

$\displaystyle \Rightarrow 52x+37=180 \ or\ x= \frac{11}{4}$

$\displaystyle \\$

$\displaystyle \text{Question 8: } \frac{(x+5)}{6} - \frac{(x+1)}{9} = \frac{(x+3)}{4}$

$\displaystyle \frac{(x+5)}{6} - \frac{(x+1)}{9} = \frac{(x+3)}{4}$

$\displaystyle \Rightarrow 6(x+5)-4(x+1)=9(x+3)$

$\displaystyle \Rightarrow 6x+30-4x-4=9x+27$

$\displaystyle \Rightarrow 2x+26=9x+27$

$\displaystyle \Rightarrow 7x=-1$

$\displaystyle \Rightarrow x=- \frac{1}{7}$

$\displaystyle \\$

$\displaystyle \text{ Question 9: } \frac{2-9x}{17-4x}=\frac{4}{5}$

$\displaystyle \frac{2-9x}{17-4x}=\frac{4}{5}$

$\displaystyle \Rightarrow 5(2-9x)=4(17-4x)$

$\displaystyle \Rightarrow 10-45x=68-16x$

$\displaystyle \Rightarrow 29x=-58$

$\displaystyle \Rightarrow x=-2$

$\displaystyle \\$

$\displaystyle \text{Question 10: } \frac{2x-3}{3x-1} = \frac{2x+3}{3x+4}$

$\displaystyle \frac{2x-3}{3x-1} = \frac{2x+3}{3x+4}$

$\displaystyle \Rightarrow (2x-3)(3x+4)=(3x-1)(2x+3)$

$\displaystyle \Rightarrow 6x^2-9x+8x-12=6x^2-2x+9x-3$

$\displaystyle \Rightarrow -x-12=7x-3$

$\displaystyle \Rightarrow 8x=-9$

$\displaystyle \Rightarrow x=- \frac{9}{8}$

$\displaystyle \\$

$\displaystyle \text{Question 11: } \frac{2-9x}{16+5x} =0$

$\displaystyle \frac{2-9x}{16+5x} =0$

$\displaystyle \Rightarrow 2-9x=0$

$\displaystyle \Rightarrow x= \frac{2}{9}$

$\displaystyle \\$

$\displaystyle \text{Question 12: } \frac{0.5x+4}{1.2x+6} = \frac{5}{3}$

$\displaystyle \frac{0.5x+4}{1.2x+6} = \frac{5}{3}$

$\displaystyle \Rightarrow 3(0.5x+4)=5(1.2x+6)$

$\displaystyle \Rightarrow 1.5x+12=6x+30$

$\displaystyle \Rightarrow 4.5x=-18$

$\displaystyle \Rightarrow x=-4$

$\displaystyle \\$

$\displaystyle \text{Question 13: } \frac{3}{2x-1} + \frac{4}{2x+1} = \frac{7}{2x}$

$\displaystyle \frac{3}{2x-1} + \frac{4}{2x+1} = \frac{7}{2x}$

$\displaystyle \Rightarrow \frac{3(2x+1)+4(2x-1)}{(2x-1)(2x+1)} = \frac{7}{2x}$

$\displaystyle \Rightarrow \frac{6x+3+8x-4}{4x^2-1} = \frac{7}{2x}$

$\displaystyle \Rightarrow 12x^2+6x+16x^2-8x=28x^2-7$

$\displaystyle \Rightarrow -2x=-7$

$\displaystyle \Rightarrow x= \frac{7}{2}$

$\displaystyle \\$

$\displaystyle \text{Question 14: } \frac{3}{x-2} - \frac{2}{x-3} = \frac{4}{x-3} - \frac{3}{x-1}$

$\displaystyle \frac{3}{x-2} - \frac{2}{x-3} = \frac{4}{x-3} - \frac{3}{x-1}$

$\displaystyle \Rightarrow \frac{x-5}{x-2} = \frac{x+5}{x-1}$

$\displaystyle \Rightarrow (x-5)(x-1)=(x-2)(x+5)$

$\displaystyle \Rightarrow x^2-5x-x+5=x^2-2x+5x-10$

$\displaystyle \Rightarrow -6x+5=3x-10 \ or\ x=\frac{15}{9}=\frac{5}{3}$

$\displaystyle \\$

$\displaystyle \text{Question 15: } (x+3)(x-3)-x(x+5)=6$

$\displaystyle (x+3)(x-3)-x(x+5)=6$

$\displaystyle \Rightarrow x^2-9-x^2-5x=6$

$\displaystyle \Rightarrow -9-5x=6$

$\displaystyle \Rightarrow 5x=-15$

$\displaystyle \Rightarrow x=-3$

$\displaystyle \\$

$\displaystyle \text{Question 16: } x(2x+3)-2x(x-5)=26$

$\displaystyle x(2x+3)-2x(x-5)=26$

$\displaystyle \Rightarrow 2x^2+3x-2x^2+10x=26$

$\displaystyle \Rightarrow 13x=26$

$\displaystyle \Rightarrow x=2$

$\displaystyle \\$

$\displaystyle \text{Question 17: } \frac{x+6}{4} - \frac{5x-4}{8} + \frac{x-3}{5} =0$

$\displaystyle \frac{x+6}{4} - \frac{5x-4}{8} + \frac{x-3}{5} =0$

$\displaystyle \Rightarrow 10(x+6)-5(5x-4)+8(x-3)=0$

$\displaystyle \Rightarrow 10x+60-25x+20+8x-24=0$

$\displaystyle \Rightarrow 7x=56$

$\displaystyle \Rightarrow x=8$

$\displaystyle \\$

$\displaystyle \text{Question 18: } \frac{3}{4} (7x-1)- \Big( 2x- \frac{1-x}{2} \Big)=x+ \frac{3}{2}$

$\displaystyle \frac{3}{4} (7x-1)- \Big( 2x- \frac{1-x}{2} \Big)=x+ \frac{3}{2}$
$\displaystyle \Rightarrow 3(7x-1)-(8x-2+2x)=4x+6$
$\displaystyle \Rightarrow 21x-3-8x+2-2x=4x+6$
$\displaystyle \Rightarrow 11x-1=4x+6$
$\displaystyle \Rightarrow 7x=7$
$\displaystyle \Rightarrow x=1$