[Technology Exercise]

a. Gra...

Question

[Technology Exercise]

a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which

3/(x − 1) < 2/(x + 1)

b. Confirm your findings in part (a) algebraically.

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the x values for which 5 divided by X minus 5 is less than 4 divided by x + 5 by graphing the functions F of X is equal to 5 divided by X minus 5, and g of X is equal to 4 divided by x + 5 together. Awesome. So it appears for this particular problem, we're asked to solve for two separate answers. Our first answer that we're ultimately trying to solve for is we're trying to create a graph of F of X and GFX on the same coordinate plane, so we're going to be graphing all of our curves on the same graph. So we're trying to plot a curve for F of X, and we're trying to plot a curve for G of X. That's our first answer. And then our second answer that we're ultimately trying to solve for is we're trying to figure out the interval notation to figure out the X values for which this inequality statement is met. So we're trying to make sure we're trying to figure out the interval. So we need to write our final answer in interval notation, but we're trying to find the interval for which these X values appease this inequality of 5 divided by X minus 5 is less than 4 divided by X plus 5. So we're trying to figure out what X values prove this inequality. So with that said, Now that we know that we're ultimately trying to solve for two separate answers, our first step that we need to take in order to solve for this particular problem is we need to find the X values for which, once again, that 5 divided by X minus 5 is less than 4 divided by X plus 5, and we will ultimately need to graph the functions for F of X is equal to 5 divided by X minus 5, and G of X is equal to 4 divided by X + 5. Fantastic. And we need to analyze where F of X is less than G of X. Fantastic. So, in order to do that, our first official step to solve this problem is we need to graph F of X is equal to 5 divided by X minus 5. And I wrote our function for F of X in red because we're going to graph our curve in red, so we'll be able to tell them apart. So let's graph our graph for GFX in green, and we'll graph our curve for F of X in red. So now we're trying to graph our function for F of X. So we need to note just by looking at the this particular function on its own, we need to be able to recall and recognize that this is a hyperbola with a vertical asymptote at X equals 5, where the function will be. Undefined, and there will be a horizontal asymptote at Y equals 0. So we need to note that when X is greater than 5, the function will decrease as X increases, and for when X is less than 5, that will mean the function will also decrease as X becomes negative. So that will mean our key points will be that X equals 6. So when X is equal to 6, that will mean that F of 6 will be equal to 5 divided by 6 minus 5, which will give us an answer of 5 when we perform that calculation. And when X is equal to 4, we will find that F of 4. So once again, we're plugging in a value of 4 and for X. So instead of rewriting the equation every single time I'm just gonna give you the evaluated answer. So when X is equal to 4, F4 is going to be equal to -5, and when X is equal to 0, that will mean that F of 0 is going to be equal to -1. Fantastic. So now, our 2nd step, moving right along here, our 2nd step now is we're gonna be trying to graph. Our function of G of X, so G of X once again is equal to 4 divided by X plus 5, and we're going to be graphing this curve in green. So once again, we need to recall and recognize the fact that our function for GFX is a hyperbola with a vertical asymptote at X. equals -5, and this is where the function will be undefined, and it also has a horizontal asymptote at Y equals 0. So we need to note that for when X is greater than -5, that means the function will decrease as X increases. And when X is less than -5, that will mean that the function also decreases as X becomes more negative. So our key points, so X is equal to 0, so when X is equal to 0, G of 0 will be equal to 4 divided by 0 + 5, which when we evaluate that, we'll get 4 5s. And similarly, when X is equal to -4, that will mean that G of -4 will be equal to 4. Once again, we're just plugging in a value of -4 and for X. So instead of rewriting the equation every single time, I'm just going to give you the evaluated answer. And when X is equal to -6, that will mean that G of -6 is going to be equal to -4. Fantastic. So, moving right along here. So now our third step is that we need to switch gears for a little for a moment here, and we need to find the intersection points. So, as we should recall a note, in order to find where F of X. is equal to G of X, we need to solve. For the following. So we need to solve for when F of X is equal to G of X to help us figure out the intersection points. So with that in mind, we need to plug in our known values for F of X and G of X. So we need to take 5 divided by X minus 5 is equal to 4 divided by X + 5. Fantastic. So, ultimately, what we need to do in order to solve for this particular problem is we need to start by cross multiplying. So we need to take 5 multiplied by X plus 5 in parentheses is equal to 4, multiplied by X minus 5. And then we need to expand our answer to get 5 X plus 25 is equal to 4 x minus 20, and we need to now rearrange our equation to isolate and solve for X. So I'm going to skip a few steps, but using a little bit of algebra, when you rearrange our expression to isolate sulfur X, you'll find that X is equal to -45. Fantastic. So that means that the graph will intersect at X equals -45. So for our fourth step now, we need to analyze our inequality of 5 divided by X minus 5 is less than 4 divided by X + 5. So from our graph, we need to note that when X is less than -45, that will mean that F of X is going to be less than G of X. Fantastic. And for when -45 is less than X and X is less than -5, that will mean that F of X is going to be greater than G of X. And also, moving right along here, it's important to note that Or -5 is less than X and X is less than 5, that will mean that F of X is going to be drumroll, less than G of X, and finally, for when X is greater than 5, that will mean that F of X is greater than G of X. Fantastic. So now, with that said, We can now solve for our 5th and final step, which is the solution. So, we need to note that the inequality, which, as we stated, for 5 divided by X minus 5. So we're gonna say 5 divided by X minus 5 is less than 4 divided by X plus 5 holds for the condition where X is less than 45, or when negative 5 is less than X, which is less than 5. So, -5 is less than X and X is less than 5. And writing this in interval notation, it will be negative infinity, so it's gonna be parentheses negative infinity -45-55. And that's it. That is our final set of answers. So now, using our information that we've gathered together. Quickly making sure to box our final answer for the interval notation, because that's one of our final answers. Let's plot our graphs for both of our curves. So when we do that, so essentially you can use whatever graphing software makes the most sense to you, but I've gone ahead and plugged both of our equations for FFX and GFX. Or rather functions for F of X and G of X into my graphing software and I produced the following set of curves. So once again, our graph for F of X is represented in red and our graph for G of X is represented in green. And as you can see, we have our different points that we found together since we plot and then we plot our different points on our curve. And as you can see, they're both hyperbola. Awesome. And once again, to quickly recap our interval notation for the values of X for which the inequality holds for will be parentheses negative infinity 45 in parentheses union 555, and that is. One of our final answers, so that's it. So as you can see, we've officially solved for our problem. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

[Technology Exercise]

a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which

3/(x − 1) < 2/(x + 1)

b. Confirm your findings in part (a) algebraically.

Key Concepts

Graphing Rational Functions

Inequalities of Rational Functions

Algebraic Confirmation of Inequalities

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