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.

1 - (x + 3) ≥ 4 - 2x

Accepted Answer

Hello today we're going to be solving the following inequality and graphing its solution and writing the solution interval notation. So the inequality given to us is two minus the quantity X plus eight. It's greater than or equal to 6 -4 x. Now in order to simplify this inequality, the first thing we want to do is distribute negative one into the quantity X plus eight. Doing so is going to give us two minus x minus eight is greater than equal to six minus four X. On the left hand side. We can go ahead and combine two and negative eight to give us negative six. So the left hand side of the inequality becomes negative x minus six. Now what we want to go ahead and do is move all the X terms to one side of the inequality and all the constants to the other. So what we can go ahead and do is add four X to both sides of the inequality and add six to both sides. And doing this is going to give us three, X is greater than or equal to 12. Now we want to go ahead and get X by itself. So what we can go ahead and do is divide both sides of this inequality by three and that's going to give us X is greater than equal to four. Now, what we want to go ahead and do is graph this inequality. So let's first start by drawing a number line and we're going to put four in the center of the number line, what this inequality is saying is that any number X can be greater than or equal to four. So since X can be greater than or four, we want all the values that go to the right of four on the number line. Or in other words we want all the values that approach positive infinity and X can be equal to four. So we can include four in our solution set and we can represent that with a solid dot on the number line. So since we want four and all the numbers that come after four, we're going to shade in the right hand side of this number line and that's going to be the graph of the inequality. Now let's go ahead and represent this graph in interval notation. So we first start by writing our lowest number of the solution set, which in this case is going to be four. Since we are including four in the solution set, we're going to put a bracket next to four and we want all the numbers that go after four. So all the numbers that go to infinity and we're going to put a parentheses next to infinity. And this is going to be the interval notation representation of the line graph. So the interval notation is bracket for common infinity parentheses and the graph that accurately represents this is going to be D. So d is the solution to this problem. So I hope this video helps you in understanding how to do this 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. 1 - (x + 3) ≥ 4 - 2x