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. -(2/3)x-(1/6)x+(2/3)(x+1)≤4/3

Accepted Answer

Hey everyone in this problem, we are asked to find the solution set for the following inequality were told to express the answer in interval notation. And the inequality were given is minus 1/4 X minus 3/8 X plus 7/12 X minus two is less than or equal to +56. Alright, so just rewriting what we're given now when we have an inequality like this with X terms and constance, what we wanna do is we want to try to get all of the X terms on one side, All of the constant terms on the other side. Okay. And then isolate for X. It's the same thing we would do if we were solving for X with an equal sign. Now we have an inequality, we're gonna do the same thing isolate X. So let's expand the left hand side first we get minus 1/4 x minus 3/ X. What's 7/12? X minus 14/12 is less than or equal to 5/6. Now we have denominators here of 4, 8, 12 and six. Let's try to determine the lowest common denominator that we can have in order to simplify this problem and you'll notice that 24 is the lowest common denominator. Okay, 48, 12 and six all divide 24 evenly. And so we want to get all of these in terms of a denominator of 24 So that we can simplify and combine the terms. So -1/4 X. Well this is going to be equal to minus 6/24 X. Since four times six is 24. We know that eight times three is 24. So we need to multiply this second term by three. So we get that nine x. Over 24. Okay the next term we have a denominator of 12 we need to multiply by two to get a denominator of 24. We have to do that to the numerator and the denominator. And so we get Over 24 x. And again in this term we have a denominator of 12. We need to multiply numerator and denominator by two. Then we get 28/24. And so on the left hand side we have negative 6/24 X minus nine X. Over 24 plus 14/24 X minus 28/24. And on the right hand side, 5/6 we need to multiply by four to get a denominator of 24. And so we get 20/24. Now that everything has the same denominator we can combine terms let's try to get all of the X. Terms on the left hand side and all of the number terms on the right hand side. So we have negative six minus nine gives us negative 15 plus 14 X. So negative 15 X plus 14 X. Is going to give us negative X over 24. And on the right hand side we have 20 plus 28 is going to be 48/24. Now we can divide or multiply each side by to remove that denominator we get negative X is less than or equal to 48. Okay. And the last step is to multiply by negative one. Now you have to be careful here. When we multiply by negative one, we need to flip the inequality. So negative X is less than or equal to 48 becomes X is greater than or equal to negative 48. Okay. Always be careful when you're multiplying or dividing by negative when you're dealing with inequalities. Now we've got our solution the solution set is going to be all of the X values that are greater than or equal to -48. Now we're asked to put this into interval notation. So how are we going to write this? Well, we know that our interval is going to go from negative 48 Up to infinity because we're talking about X values that are greater than -48. We know that we never include infinity. So we have a round bracket to indicate that we don't include that value. Now -48. Well this is X is greater than or equal to negative 48. So we use a square brackets to indicate that we include negative 48 and that is the interval that represents our solution set. And so if we look at our answer choices, we see that this corresponds with answer choice D. Thanks everyone for watching. I hope this video helped you in the next one.

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Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. -(2/3)x-(1/6)x+(2/3)(x+1)≤4/3