Question 1207383
i graphed the equation and it looks like you have a vertical asymptote at x = -3.


if so, it means that x approaches -3 but never touches it.


i don't believe you will actually be able to solve the original equation at x = -3.


the value of y at x = -3 should be undefined.


since your equation is y = 5/(x+3, the value of y is undefined at x = -3 because the divisor becomes 0.


same for (8+x)/(x+3)


if you multiply both sides of the equation by (x+3), you eliminate the division by 0 as a possible solution, but that isn't the original equation.


the graph clearly shows that there is a vertical asymptote at x = -3.


if you do the calculations on the origiinal expressions on each side of the equation, you will see that y gets more positive as x approaches -3, or more negative as x approaches -3 from the other direction, but you can never get x = -3 as a possible soluton because of the division by 0.


here's my graph.


<img src = "http://theo.x10hosting.com/2024/053101.jpg">


here's a reference on asymptotes.


<a href = "https://www.mathsisfun.com/algebra/asymptote.html" target = "_blank">https://www.mathsisfun.com/algebra/asymptote.html</a>


the key to finding the asymptotes is that you have to look at the original equation.


if you change the equation by multiplying both sides by the denominator, you lose sight of the fact that there are vertical asymptotes.


i think that's what you did when you multiplied both sides of the equation by (x+3).


using graphing software helps you to see the equation to determine if any of those asymptotes exist.


the reference i gave you discusses ways to determine them, i believe.


the graphing software i used is at <a href = "https://www.desmos.com/calculator" target = "_blank">https://www.desmos.com/calculator</a>


here's another reference on asymptotes that may be more informative.


<a href = "https://www.andrews.edu/~rwright/Precalculus-RLW/Text/02-07.html#:~:text=To%20find%20the%20vertical%20asymptotes,zero%20and%20solve%20for%20x.&text=This%20is%20already%20factored%2C%20so,factor%20to%20zero%20and%20solve.&text=Since%20the%20asymptotes%20are%20lines,%3D%203%20and%20x%20%3D%201." target = "_blank">https://www.andrews.edu/~rwright/Precalculus-RLW/Text/02-07.html#:~:text=To%20find%20the%20vertical%20asymptotes,zero%20and%20solve%20for%20x.&text=This%20is%20already%20factored%2C%20so,factor%20to%20zero%20and%20solve.&text=Since%20the%20asymptotes%20are%20lines,%3D%203%20and%20x%20%3D%201.</a>