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) showing regions where the inequality is satisfied.

Accepted Answer

Hey, everyone in this problem, we're asked to find the solution for the following inequality. And the inequality we're given is three, X multiplied by X minus three, multiplied by X minus four is greater than, or equal to zero. Using the graph of F of X equals three, X multiplied by X minus three, multiplied by X minus four, which is given below. And we're asked to express the answer in interval notation. So we have the graph of F FX given here in red. And we're giving four answer choices as well. Each of them being the union of two different intervals. And we're gonna come back to each of these answer choices as we work through the problem. Let's go back up where we can see our graph and we can see our inequality. We're gonna start by writing out this inequality. We have three X multiplied by X minus three, multiplied by X minus four is greater than or equal to zero when we're given an inequality. And we're asked to solve it using a graph. What we wanna do is figure out how is this graph related to this inequality? OK. So the graph of F of X were given, what you'll notice is that the left hand side of this inequality is F of X, OK. Three X multiplied by X minus three, multiplied by X minus four. And so what we have in our inequality is actually F FX is greater than, or equal to zero. Now, F of X is what we have graphed here. So if we want F of X greater than or equal to zero, we called it, the XX values are represented along the X axis and the F of X values are represented by the Y axis. So we wanna find where the Y values are greater than, or equal to zero, which means where this graph is above the X axis. So let's take a blue color here and we're gonna trace this graph everywhere where F of X or that Y value is greater than, or equal to zero. It's gonna start at zero and this function increases up to a maximum of around 18 before it decreases. Again, it's gonna hit the X axis at X equals three. It's gonna continue down negatively. And we're not gonna color that blue because it's not gonna be included in our interval. Those are F FX values that are less than zero at X equals four. This crosses the X axis again, it becomes positive. And so the remaining part of this function is going to be drawn in blue as well. So we've drawn out our two intervals. And what we wanna do is figure out the X values that give us these F of X values. So for the first interval, these positive values start at X equals zero can go up to X equals three. Now, I haven't included brackets yet because we need to figure out if they should be square brackets or round brackets. Now, our inequality says that F FX is greater than, or equal to zero. So our function can lie on the X axis or above it at X equals zero. The function F of X is equal to zero. OK. That's OK. That satisfies our inequality. So we use a square bracket at zero to indicate that that's included. The same thing happens at three at X equals three. Our function F of X is equal to zero. OK? And you can always check with the equation as well. OK? We were given the equation for this graph. If you substitute X equals zero or X equals three, you get that, that function is equal to zero. And that's OK. It satisfies our inequality. So we use square brackets for both of those intervals. OK. So that's our first interval. If we take a look at our answer choices, we can already eliminate two of the options. We can eliminate option B because it has the first interval going from 0 to 4 instead of 0 to 3. And we can eliminate option D because it uses a round bracket at zero instead of a square bracket. OK. So it's excluding zero when it should be included. Now, continuing with our graph, we wanna look at the next interval, we're gonna take the union of that next interval and the next interval where we have positive values starts at four. And we're gonna use a square bracket there again because at X equals four, the function F of X is equal to zero, which satisfies our greater than or equal to inequality. And the function is gonna continue to be positive for the remaining X values. OK. As X gets larger and larger F of X is getting more and more positive, larger and larger in the positive direction. And so this interval is gonna go all the way up to infinity and we always use a round bracket at infinity. And so that's the final answer. Those are those two intervals that make up the solution. In this case, we have the interval from 0 to 3 including end points union with the interval from four to infinity where we include the end point of four. Going back to our answer choices. This corresponds with answer choice A, OK. So answer choice A is a correct interval. Answer choice. C had the wrong interval for that second interval. It had it from 3 to 4 instead of from four to infinity. That's it for this one. Thanks everyone for watching. I hope this video helped

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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