In Exercises 95–102, use interval notation to represent all value...

Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y = 2x - 11 + 3(x + 2) and y is at most 0

Accepted Answer

Hello everyone in this video. We're gonna be looking at this practice problem where we want to find the values of X that satisfy the condition that why is at most five. And in this case we're given that Y. Is equal to three X plus parentheses one minus X. Parentheses minus two. We want to make sure to express that range in interval notation. And in this case let's translate why is that most five into sort of inequality So we can solve it. So why is that most five is telling me that five is the upper bound of values that Y could be and there doesn't really seem to be a lower bound. So I'm just going to call that negative infinity. So now that I have it written like this it's easier for me to translate into an inequality or I can see that I want why to be less than or equal to five. So now I have this inequality that represents the range of Y values that I want to try to find. But remember that the problem is asking for the range of X values that satisfy this condition. So in this case we're given what Y. Is equal to Where Y is equal to three x plus one minus x minus two. I'm going to go ahead and simplify this where I have three X plus one minus X minus two. I combine like terms I'm left with three x minus x two X and then positive one and negative two. I'm left with negative one. So why is equal to two X -1. So I'm gonna go ahead and Subsequute in two X -1 into this. Y. So now I'm gonna have two X -1 is less than or equal to five. And now you can see that I have an X. In my inequality. So this is going to give me a range of X values that will result in why being at most five or why being less than or equal to five. And if I solve for X. And this inequality I have two X less than or equal to six divided by two. I have X less than or equal to three. So now I have a range of X values that will satisfy why being less than or equal to five. If I translate this into interval notation you can see that I have an equal to so it's going to be a bracket and three seems to be my upper limit so I'll write three and there doesn't seem to be a lower limit. So I'll say negative infinity. And this becomes my range for X. Values that satisfy the condition of why being less than or equal to five. And you can see that this aligns with answer choice A. So hopefully this video hoped and see you next time

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