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 a solution for the following inequality 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 the inequality we're given is three, X multiplied by X minus three, multiplied by X minus four is greater than zero. And we're showing the graph of F FX drawn in red. Now we're also given four answer choices. OK. Each of them containing different intervals. So options A B and C have the union of two different intervals. And option D is a single interval. So we're gonna come back to those as we work through this problem, let's go back up to our interval and get started. OK? So we have our in inequality three X multiplied by X minus three, multiplied by X minus four is greater than zero. Now, when we have an inequality and we're asked to solve it using a graph, what we wanna do is figure out how that graph relates to our inequality. Now, in this case, you'll notice that the left hand side of our inequality is F of X K three, X multiplied by X minus three, multiplied by X minus four is F of X. And so what we really have here is that F of X is greater than zero. Now, our graph shows F of X, we want F of X greater than zero. So remember that on our graph, the F of X value is represented in the, by the Y axis. OK. So what we want is a Y value that is greater than zero. Where do we get that? Will we get that everywhere where the function is above the X axis? So if we take and we draw everywhere where this function is above the X axis, this starts just after X equals zero. The function increases up to a maximum around 18 before decreasing again. OK? That stops at X equals three K at X equals three. The function is equal to zero. It dips down and becomes negative until X equals four. And then after X equals four is gonna be positive again. So in blue, now we've drawn all of these areas where the function F of X is greater than zero. OK? Where it's pause and what we wanna do is write these intervals, the X values that give us these positive F of X values in interval notation. So for the first interval, it starts at X equals zero and goes until X equals three we have from 0 to 3. And we have to figure out if this includes the end points or not. Do we want brown brackets or square brackets? OK. Well, our inequality is strictly greater than zero. OK. We cannot have F of X equal to zero. So at X equals zero or at X equals three, the function F of X is equal to zero. And we can see that from the graph and we can also see that from the equation of that function. Yeah, we wanna in exclude those points because F of X equals zero is not a solution. So we have round brackets there to indicate that we're excluding those points. Now we're gonna take the union of that second interval. Before we do that, let's check up on our answer traces and see if there's anything we can eliminate already. So there are two things we can eliminate. The first one is option B OK? Because it has this first interval going from 0 to 4 instead of 0 to 3. And the second thing we can eliminate is option D because it only has one interval in the solution. And we know from the drawing on the graph that we have to have two intervals. OK? So option B and option D are eliminated, we're left with option A or C, OK. Back to our graph. Back to our inequalities. The second interval starts at X equals four and is gonna go all the way up to positive infinity. OK. For every X value after four, this function is just getting larger and larger in the positive direction. And so that's going to satisfy that inequality of F FX is greater than zero. Now, for infinity, we always in use a round bracket and we can't include that value for four. It's the exact same reasoning as before. OK. When X is equal to four, the function F of X is equal to zero, which does not satisfy your inequality. So we use a round bracket to indicate that we exclude that value. And so the complete solution is gonna be the union of the interval from 0 to 3 with the interval from four to infinity where none of the end points are included. If we compare this to our answer choices, we see that this corresponds with answer choice. A OK, answer choice C is the wrong choice because that second interval goes from three to infinity instead of from four to infinity. 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 Inequalities

Roots and Zeros of a Polynomial

Using Graphs to Solve Inequalities

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