Use the graph to solve each equation or inequality. Use interval ...

Question

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(X-2) / {(X-1)(X-3)} ≤ 0

Graph of the function 2(X-2) / ((X-1)(X-3)) showing regions for the inequality.

Accepted Answer

Hey, everyone in this problem, we're asked to find the solution for the following inequality which were given as three X minus five divided by X minus four, multiplied by X minus six, less than are equal to zero. We want to solve that using the graph of F FX equals three, multiplied by X minus five, divided by X minus four, multiplied by X minus six. That's given below. And we're asked to express the answer in interval notation. So we're given the graph of the function F of X it's drawn in red. And then we're given four answer choices in all four of the answer choices have very similar intervals. They have the union of the interval from negative infinity to four with the interval from 5 to 6. And they just differ in whether they have square or round brackets for the, in, for the end points to indicate whether it's included or not included. So let's go back up to our inequality. OK. So we have three multiplied by X minus five. All of that is divided by X minus four multiplied by X minus six. And we want, when this is less than, or equal to zero. Now, the graph of the function we're given this function F of X is exactly what we have on the left hand side of our in equally OK. Three multiplied by X minus five, divided by X minus four, multiplied by X minus six. So what we really want is to find where F FX is less than or equal to zero. Now, if we look at our graph K F FX is represented by the Y axis. So what we want is we want all of the X values such that this graph is below Y equals zero or on it, which is everything below or on the X axis. OK. So we can see the function I am going to trace in blue. This is below the X axis and this is for X values as far left as our graph grew goes down to almost four. Then the graph is positive X equals five. We have an X intercept. OK. And then it continues to decrease downwards until around six before it jumps back positive. OK. So the product I've traced in blue is the part where our function F of X is less than or equal to zero. And again, we want to find the associated X values. So let's start writing the first interval. OK. This first part where we have negative values and it's gonna continue as X goes to negative infinity, we can see we're approaching the X axis and So we're gonna go from negative infinity up to, yeah, we can see we have almost like a vertical ascent to at X equals four. OK. And that four is where it switches. Now, the question is, do we include for or do we not include it? OK. This end point now at X equals four, this function is not defined. OK. Again, there's a vertical a tope here. And so we are going to exclude four. So we're gonna use a round bracket. We're gonna have a union of the next interval and the next time this function becomes zero or negative is that X equals five. So we're gonna have the union from five and we're gonna use a square bracket here. OK? Because we want, when the function is less than or equal to zero and at X equals five, that function is equal to zero. So we use a square bracket to include five and this interval is gonna go up until six. Hey, at six, it switches from being a negative back to being positive and just like at X equals four, this is a vertical axum to at X equals six. The function is not defined for that value. And so we do not include that point. So we use a round bracket at six. OK? And everything bigger than six for all of our X values bigger than six, this function is positive. It's above the Y axis which does not satisfy our inequality. So the interval we've written is the entire solution. We have the interval from negative infinity up to four where four is not included. And then we take the union of that with the interval from 5 to 6 where five is included and six is not. If we compare this with our answer choices, we can see that this corresponds with answer choice. A, the answer choice B includes the end 0.4 which we know is incorrect answer. Choice C does not include the end 0.5. OK. Which we also know is incorrect because at five, our function is equal to zero, which satisfies the condition. And option D includes the M 0.6 which again we know is incorrect. Visa function is not defined now. So the correct answer is option A. Thanks everyone for watching. I hope this video helped see you in the next one.

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(X-2) / {(X-1)(X-3)} ≤ 0

Key Concepts

Rational Functions

Inequalities and Interval Notation

Graph Analysis

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