1. Solve the rational equation quantity 4 times x plus 3 end quantity divided by 5 equals quantity 8 times x minus 1 end quantity divided by 9. x = 0.5 x = 2 x = 8 x = 92. Determine the vertical asymptote for the rational function f of x equals quantity x minus 4 end quantity divided by quantity 2 times x plus 3 end quantity. x = −4 x equals negative three halves x equals three halves x = 4

Answer:1. x = 82. [tex]x=-\dfrac{3}{2}[/tex] Step-by-step explanation:1. Solve the rational equation [tex]\dfrac{4x+3}{5}=\dfrac{8x-1}{9}[/tex]First, cross multiply:[tex]9(4x+3)=5(8x-1)[/tex]Now, use distributive property:[tex]36x+27=40x-5[/tex]Separate terms with x and without x into different sides of equation:[tex]36x-40x=-5-27[/tex]Simplify:[tex]-4x=-32[/tex]Divide by -4:[tex]x=8[/tex]2. The rational function[tex]f(x)=\dfrac{x-4}{2x+3}[/tex]This rational function is undefined for all values of x, for which the denominator is equal to 0. Find these values:[tex]2x+3=0\\ \\2x=-3\\ \\x=-\dfrac{3}{2}[/tex]This means that the line [tex]x=-\dfrac{3}{2}[/tex] is a vertical asymptote for the rational function f(x).