Question

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 = 9

2. 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

Answer:

1. x = 8

2. $x=-\dfrac{3}{2}$

Step-by-step explanation:

1. Solve the rational equation

$\dfrac{4x+3}{5}=\dfrac{8x-1}{9}$

First, cross multiply:

$9(4x+3)=5(8x-1)$

Now, use distributive property:

$36x+27=40x-5$

Separate terms with x and without x into different sides of equation:

$36x-40x=-5-27$

Simplify:

$-4x=-32$

Divide by -4:

$x=8$

2. The rational function

$f(x)=\dfrac{x-4}{2x+3}$

This rational function is undefined for all values of x, for which the denominator is equal to 0. Find these values:

$2x+3=0\\ \\2x=-3\\ \\x=-\dfrac{3}{2}$

This means that the line $x=-\dfrac{3}{2}$ is a vertical asymptote for the rational function f(x).