Solve each inequality. Give the solution set in interval notation...

Question

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (4x+7)/-3≤2x+5

Accepted Answer

Hey, everyone. Welcome back in this problem. We are asked to find the solution set for the following inequality. We're told to express the answer in interval notation. And the expression the inequality we're given is nine X plus two divided by negative five is less than or equal to three X plus two. So let's rewrite what we were given nine X plus two over negative five is less than or equal to three X plus two. Yeah, when we have an inequality and we're asked to find the solution set. What that's asking us to do is find all of the X values that satisfy this inequality. And in order to do that, we want to isolate X. Okay. So we're going to isolate it just like we would with an equation with an equality. Okay with an equal sign, we're gonna get all the X terms to one side, all of the constant terms to the other side and then isolate X. So let's start here by getting rid of this negative five. Now we can multiply both sides of the equation by negative five. On the left hand side, we'll have nine X plus two on the right hand side, we're gonna have three X plus two times negative five. Now I've left out the inequality because I want to make this point clear we have multiplied by a negative. When you're dealing with inequalities. If you multiply or divide by a negative, you need to flip the inequality. Okay. So now this becomes greater than or equal to instead of less than or equal to. So we have nine X plus two is greater than or equal to three X plus two times negative five. Alright? And you can think of this in another way if we want to write a little aside here. Mhm We can write this as nine X plus two over negative one times five is less than or equal to three X plus two. We can multiply by the positive five. OK? Nine X plus two over negative one is less than or equal to five times three X plus two. And when we apply the negative, we get negative nine X minus two is less than or equal to five times three X plus two. And in order to rearrange this to be equal to this one, we would need to move everything to the opposite side. So the five times three X plus two, we need to move to the left, it's positive. So when we move it to the left, it becomes negative. So we get negative five times three X plus two, we have negative nine X minus two, we want to move it to the right hand side. And in order to do that, it's going to become positive nine X plus two. Okay? And so this inequality is equal to what we have here. Okay? And so that's the reasoning behind why we have to flip that sign when we multiply by a negative. Alright, so always be careful there. Now we can keep going. Let's expand on the right hand side, we have nine X plus two is greater than or equal to negative 15 X minus 10. Let's move all the extras to the left hand side and all the constant terms to the right hand side, we get nine X plus 15 X which gives us X. And on the right hand side, we have negative 10, we're gonna move this plus two over it's gonna become negative And we get -12. Last step we're going to divide by 24 here we're dividing by a positive. So we don't need to flip the sign of our inequality and we get that X is greater than or equal to negative 12/24. Okay. And this is equivalent to writing X greater than or equal to negative a half. Alright. And that's it. That is the solution set for inequality. We can have any X value that is greater than or equal to negative half and were asked to rate this in interval notation. And so the lowest value we have is going to be negative a half. And we have a square bracket here to indicate that we're including negative half because we have greater than, or equal to and we have this equal peace. So we want X bigger than negative a half. And so we have negative half all the way up to positive infinity or are possible X values. And we always use a round bracket on infinity because we don't include rate at infinity. Alright, so that is the interval that represents our solution said, if we compare that to the answer choices, we see that that's going to correspond with answer choice B so we have the interval from negative one half including negative half all the way up to positive infinity. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (4x+7)/-3≤2x+5

Key Concepts

Solving Linear Inequalities

Handling Inequalities with Fractions

Interval Notation for Solution Sets

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