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

Question

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 7x(x - 1)(x - 2) < 0

Graph of the function 7x(x - 1)(x - 2) for solving linear inequalities.

Accepted Answer

Hey, everyone in this problem, we're asked to find a solution for the following inequality using the graph of F FX equals three, X multiplied by X minus three, multiplied by X minus four, which is given below in case we have the graph shown in red. And we're asked to express the answer in interval notation. Now, the inequality we're given is three, X multiplied by X minus three, multiplied by X minus four is less than zero. We're given four answer choices, options A through D and all of them have the union of two different intervals. And we're gonna come back to these answer choices as we work through this problem. So starting with our inequality, which we're gonna rewrite three, X multiplied by X minus three, multiplied by X minus four is less than zero when we have an inequality and we're asked to solve it using the graph. What we want to do is figure out how does that graph relate to our inequality. And what you'll notice is that the left hand side of our inequality three X multiplied by X minus three multiplied by X minus four is equal to F of X So what our inequality is telling us is that we have F of X less than zero. Now, F FX is what is grafted in the graph we're given that is a function graph. OK. Now let's recall that the F of X value is represented on the Y axis in our graph. So what this is telling us is we want Y values less than zero, where do we have Y value is less than zero. Well, that's gonna be below the X axis. So what we want to do is find all of the X values that give us a Y value below the X axis. Let's go to our graph. We're gonna draw in blue and we're gonna trace the function where it is less than zero. OK? Below the X axis. So anything to the left of X equals zero, OK? We can see that this function is negative. It's below the X axis, that's what we want. OK? Then it's gonna become positive. It's gonna reach some maximum before coming back down. It's gonna cross the X axis at X equals three. It's gonna be negative, then it's gonna start to increase again and it's gonna cross the X axis again at X equals four before increasing in the positive direction. OK. So these two small portions we've traced in blue are where that function is less than zero. And now we want to find the corresponding X values that make this happen. So for this leftmost interval. And we can see that for any value of X on the left-hand side, this function is going to get more and more negative. Hey, it's looks like it's going down to negative infinity. So for any X value to the left of zero, meaning negative infinity, all the way up to zero, this function is less than zero, which satisfies the inequality. Now, we use a round bracket on negative infinity because we never include exactly negative infinity on zero. We need to figure out whether we want a square bracket or a round bracket. Now, in this case, our inequality is strictly less than OK, at X equals zero. Our function F of X is equal to zero. We can see that from the graph and we can see that from the equation. If we were to substitute X equals zero in the equation, we would get F of X equals zero which does not satisfy our inequality. So we're gonna use a round bracket to indicate that that's not included. So this is our first interval. We're gonna take a look at our answer choices right now to see if we can eliminate any of them. And what you'll notice is we can actually figure out the correct answer just from this first portion. Option B and C have the first interval being from negative infinity up till three. OK, which we found is not true. So B and C can be eliminated and option D has no X values less than zero that are included. OK. Both of the intervals are above zero. That's not correct either. So we can actually eliminate all option choices except for a, OK. So we expect a to be the correct answer, but let's write down the second interval just to make sure we've made no mistakes and to get some practice doing this and we have our first interval, we're gonna take the union with the next interval in the next interval. OK? We have X values from 3 to 4. OK. That's where we've drawn blue on our graph where it's below the X axis. And again, we're using round brackets to indicate we're not including the end points because at X equals three or X equals four, the function F of X is equal to zero, which does not satisfy our inequality. And so the complete answer here is the union of the two intervals. The first being negative infinity up to zero. The second being 3 to 4 where we don't include any of the end points. And this does indeed correspond with answer choice. A like we expected. 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. 7x(x - 1)(x - 2) < 0

Key Concepts

Polynomial Functions and Their Graphs

Solving Polynomial Inequalities Using Graphs

Interval Notation

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