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

Question

Solve each inequality. Give the solution set in interval notation. (1/3)x+(2/5)x-(1/2)(x+3)≤1/10

Accepted Answer

Hey, everyone in this problem, we are asked to find the solution set for the following inequality and to express the answer and interval notation and the inequality we're given is 3/4, X plus 5/6 X minus 1/8 times X plus two is less than, or equal to 1/24. We're gonna rewrite what we were given. Now when we have an inequality like this, we're not gonna treat it any differently than if we had inequality. So if we had inequality and we were asked to solve for X, we would move all the X terms to one side, all the constant terms to the other. In order to isolate X. In this case, we're asked to find the solution set. So what we want to find is the possible X values or the X values that satisfy this inequality. So we're gonna do the same thing. We're gonna move all the extremes to one side, all of the constant terms to the other side and try to isolate for X. No, let's start by expanding this left hand side, We get 3/4 x plus 5/ X minus 1/8, X minus 2/8 is less than or equal to 1/24. Now, in order to combine these terms together, we need a common denominator right now, we have denominator of 468 and 24. And so our lowest common denominator is going to be 24. Starting on the left hand side, we have a denominator of four. In order to get a denominator of 24, we need to multiply by six, we multiply the denominator by six. We need to do the same to the numerator. And so we get 18 x over 24. Okay. The next term 5/6 X in order to get the denominator six to be 24 we need to multiply by four. We do the same to the numerator and we get that 20 X over 24. In the next term, we have eight in the dominator. In order to turn this into a 24, we need to multiply by three. We do it to the denominator. So we need to do it to the numerator as well. And so we get -3 x over 24. This final term on the left hand side also has a denominator of rates. We need to multiply by three. Again, we're gonna -6/24 and all of this is less than or equal to 1/24. Now, we have a common denominator. So we can combine these terms together when we're looking at our X terms, we have 18 X over 24 plus 20 X over 24 minus three X over 24. And so this gives us 35 x over 24. Now we're going to move this constant -6/24 to the other side. So it's going to be added just like if we were moving it within equals, we get 1/24 plus 7/24 or sorry, 1/24 plus 6/24 which gives us 7/24. We can multiply by 24 on both sides to remove those denominators. And we're going to have that 35 X is less than or equal to seven. We're going to divide by 35 to isolate X. We get that X is less than or equal to 7/35. Now, seven and 35 are both divisible by seven. So we can simplify this. This is going to be X is less than or equal to 1/5. And that's the solution. That is the X values that satisfy this equation. Now, we're asked to write this in interval notation. So let's go ahead and do that. We want x less than or equal to 1/5. So we can have values from negative infinity. Okay, we use a round bracket here because we don't include infinity. We can have values all the way up to 1/5. And we're going to use a square bracket because we have less than or equal. So our values can be equal to 1/5. And this is going to be our interval, all of the values from negative infinity, all the way up to and including 1/5, but nothing bigger. There's our interval. And if we compare this to our answer traces, we see that this is going to correspond with answer choice. C 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. (1/3)x+(2/5)x-(1/2)(x+3)≤1/10

Key Concepts

Solving Linear Inequalities

Combining Like Terms and Distributive Property

Interval Notation

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