When 4 times a number is subtracted from 5, the absolute value of...

Question

When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we're given a word problem where were given that the absolute value function of the sum of two times the number and three is at most six. And we need to express the points that satisfy that condition in interval notation. So in this case let's first look at what the problem is saying and try to translate the words into mathematical equations. So the first part is saying the absolute value function. So we know that the absolute value is just those two bars and it's saying the absolute value function of the sum. So we know that we're gonna be adding two things. We're adding two times a number and three. So it doesn't indicate the number. So I'm just going to say that a number is the same as X. So I'm going to say I'm adding two X plus three and this is what's going inside the absolute value function. And then the last part is saying that that absolute value function Is at most six. So at most six means that the six is the maximum. So my absolute value has to be less than or equal to that. So now I've written out an inequality for what the problem is saying and notice how we're working with an absolute value inequality. So we needed to keep in mind that an absolute value could have a positive number or a negative number on the inside of this but we'll come out as only positive but we still need to account for those two scenarios. So in order to solve for the negative, I'm going to pretend we have a negative in front of the inside number. So I have negative two X plus three less than or equal to six. And if I had a positive number on the inside I would have positive two X plus three less than or equal to six. And you can see that if I'm solving for my negative, I have to divide the negative across the inequality which makes it change direction. So now I have greater than or equal to and a negative six. So I'm left with two X plus three greater than or equal to negative six. And now I can continue solving for X. So I can subtract three I have two x greater than or equal to -9. If I divide by two I'm left with X greater than equal to negative nine house and I can do the same simplification for if the number was positive or I have positive two X plus three less than or equal to six. So I can subtract three and I have two X less than or equal to three, divide by two. I'm left with X less than or equal to three halves. So now you can see I've solved or simplified both of my inequalities if the number was negative or if the number was positive so I can go ahead and try to visualize these on a number line. So I'm just going to draw a number line. Not really to scale. Gonna say this is -9/2, this is zero and then I have positive 3/2. So my first interval is saying that X has to be greater than or equal to negative nine house which means I would start At the -9 house and access to be greater. So I have to go to the right into my positive numbers. My second range is saying that X has to be less than or equal to three halves. So I start at three halves and access to be less than or equal to. So I go to the left so you can see that these two intervals or ranges for X. Sort of overlap in this section between negative nine house and three halves. So that becomes the range that satisfies the condition for the absolute value of two X plus three being less than or equal to six. So I can write this in interval notation as bracket negative nine house, 23 halves and then another bracket. So this becomes my final answer. And you can see that this aligns with answer choice B. So hopefully this video helped and see you next time

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