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. 2(X-2) / {(X-1)(X-3)} < 0

Graph of a rational function illustrating linear inequalities for college algebra.

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to find the solution for the following inequality which were given as three multiplied by X minus five, divided by X minus four, multiplied by X minus six is less than zero. When we're asked to do that, using the graph of F of X equals three multiplied by X minus five, divided by X minus four multiplied by X minus six. And we're told to express the answer in interval notation. Now we're given our graph of F FX here shown in red and we're also given four answer choices, option A through D and we're gonna come back and look at those once we figure out what we think the correct intervals are based on the graph. So starting with our inequality three multiplied by X minus five, divided by X minus four, multiplied by X minus six is less than zero. Now, when we're solving an inequality using a graph, what we wanna do is figure out the relationship between the graph and the inequality. And if you look the function F of X that we're given the graph of is exactly what we have on the left hand side of our inequality. And so what this question is really asking us to do or what this inequality really shows is that F of X is less than zero. Now, on our graph F of X is represented by the Y axis. OK. So we want Y values that are less than zero. So we want to look at our graph, we wanna find Y values below the X axis and figure out what X values give us those Y values. So we look at our function in our graph, I'm gonna trace the parts of this graph that are below the X axis. And that includes the part that starts on the far left of the graph all the way up until almost four. Then our function jumps up two positive values. K continues with positive values until X equals five. It switches back over to negative values until almost six before it jumps back up to positive values for the rest of the X values. And so if we're looking at the X values where we've drawn blue over this graph where we have Y below the X axis, we can see that this starts at negative infinity. This function is approaching the X axis like a horizontal as to. And so that is gonna stay negative for all of the X values to the left. So from negative infinity up to a value of four, OK. At four, this switches up to a positive function So we go to four. Now, the question is whether we need to include the M 40.4 or not, in this case, four is a vertical ASOM tool and the function is actually not defined at four. It switches from being going really, really negative to really, really positive. And so we're gonna use a round bracket to indicate that we are not including that endpoint. Now we're gonna take the union of that interval with the next interval. The next interval starts at five and goes up to six. OK. So we go from 5 to 6 and obvious brown brackets here as well. Y well, for X equals five K, when X equals five, Y is equal to zero, our inequality wants F FX to be strictly less than zero, not less than or equal. So we cannot include that X value of five. And for six, it's very similar to the case with X equals four. OK. Six is a vertical Asymptote. The function is not defined for that value. And so we use a round bracket and that's it. We have those two intervals where the function is negative. And that is our complete answer. The union of the two comparing this to our answer traces. OK. Option A, we have the interval from negative infinity to four union with the interval from 5 to 6, which is exactly what we found. So that is gonna be the correct answer. Option B has the first interval going up to negative four instead of positive four. So that one's not correct. Option C has the first interval going from negative infinity up to five instead of four. So that's incorrect as well as having an incorrect second interval. And for option D, same thing that first interval goes up to negative six instead of four. And the second interval is incorrect as well. And so there you have it. 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 Inequalities

Critical Points and Sign Analysis

Graph Interpretation of Rational Functions

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