To determine the [tex]$x$[/tex]-intercept of the function [tex]\( y = \frac{6x - 18}{x + 9} \)[/tex], we need to find the value of [tex]\( x \)[/tex] where [tex]\( y \)[/tex] is equal to zero. In other words, we need to solve the equation [tex]\( y = 0 \)[/tex].
Here's the step-by-step process:
1. Set [tex]\( y \)[/tex] equal to zero:
[tex]\[
0 = \frac{6x - 18}{x + 9}
\][/tex]
2. To eliminate the denominator, multiply both sides of the equation by [tex]\((x + 9)\)[/tex]:
[tex]\[
0 \cdot (x + 9) = 6x - 18
\][/tex]
Since anything multiplied by zero is zero, this simplifies to:
[tex]\[
0 = 6x - 18
\][/tex]
3. Now, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[
6x - 18 = 0
\][/tex]
4. Add 18 to both sides of the equation to move the constant term to the right side:
[tex]\[
6x = 18
\][/tex]
5. Lastly, divide both sides by 6 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{18}{6}
\][/tex]
[tex]\[
x = 3
\][/tex]
Therefore, the [tex]$x$[/tex]-intercept of the function [tex]\( y = \frac{6x - 18}{x + 9} \)[/tex] is [tex]\((3, 0)\)[/tex].
