In all exercises, other than exercises with no solution, use inte...

Question

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

4(x + 1) + 2 ≥ 3x + 6

Accepted Answer

Hello today we're going to be solving the following inequality and graphing its solution and representing the solution in interval notation. So what we are given is five times the quantity X plus two plus three is greater than or equal to two X plus 13. Now, in order to start simplifying this, we're going to first distribute five into the quantity X plus two. This is going to give us five, X plus 10 plus three is greater than equal to two, X plus 13. Now we can go ahead and combine 10 and three to give us 13. So the left hand side of this inequality becomes five X plus 13. Now, what we want to go ahead and do is get all the X terms to one side of the inequality and all the non X terms to the other side. So what we can go ahead and do is subtract two X to both sides of the inequality and subtract 13 to both sides. And this is going to give us three, X is greater than or equal to zero. Finally we want to go ahead and get X by itself. So we can divide both sides of this inequality by three. And doing this is going to give us X is greater than equal to zero. So this is a simplified version of our given inequality. Let's go ahead and graph this now. So we start by drawing a number line and we're going to go ahead and put zero in the middle of the number line. This inequality is saying that we want all the numbers X to be greater than or equal to zero. So we want all the numbers that come after zero or in other words, all the numbers that approach positive infinity and X can equal to zero. So we're going to include zero in our solution set. So we can represent that on the number line as a solid dot at zero. And since we want all the numbers that come after zero, we're going to shade in the whole right hand side of the number line. And this is going to be the graph of our given inequality. Now we want to go ahead and write this graph in interval notation. So we first start off by writing our lowest number in the solution set, which in this case is going to be zero. Since we are including zero in our solution set, we're going to give it a bracket and we want all the numbers that come after zero. So all the numbers that go to infinity. So we're going to have zero comma infinity. And we're going to close it off with the parentheses. And this is going to be the interval notation for the given graph. And the option that accurately represents this solution is going to be B. So b is the solution to this problem. So, I hope this video helps me in approaching this type of inequality problem and I'll go ahead and see you all in the next video

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In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 4(x + 1) + 2 ≥ 3x + 6