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

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. x/4 - 3/2 ≤ x/2 + 1

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 X over five plus 3/ less than or equal to X over four plus one. Now we don't want to work with these fractions given to us. So what we want to go ahead and do is multiply every term by the lowest common denominator. That way we can go ahead and get rid of the denominators. The denominators given to us are five and four. So the lowest common denominator between five and four is going to be 20. So we're going to multiply every term in the given inequality by 20. And doing this is going to give us four X plus five times three is less than or equal to five X plus 20 times one. Which is 20 now, five times three is going to give us 15. So we can go ahead and rewrite this as four X plus 15. 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 side of the inequality. And in order to do this, we can go ahead and subtract four X to both sides of the inequality And subtract 20 to both sides of the inequality. And doing this is going to give us negative five is less than or equal to five X minus four eggs, which is just going to be X. So we get negative five is less than or equal to X. That's the simplified form of this inequality. Now, what we want to go ahead and do is graph this. So let's go ahead and draw a number line And we're going to put negative five in the middle. Now what this inequality is saying is that we want all numbers X to be greater than or equal to negative five. So since we want all the numbers that are greater than negative five, we want all the values starting from negative five and going to infinity. Furthermore, this inequality states that X can be equal to negative five as well. So we can go ahead and include negative five in our solution set and we can represent that with a solid dot. And again we want all the values greater than -5. So we're going to shade in the right hand side of this number line. And now we want to represent this graph an interval notation. So since negative five is the lowest number in our inequality or in our solution set, we're going to start with negative five. Since we are including negative five in the solution set, we're going to indicate a bracket next to negative five. And a bracket means that we're including a value in the solution set. And we want all the numbers going from negative five to infinity. So we're gonna have bracket negative five comma infinity and close it off with the parentheses. And that's going to be the interval notation of this number line. And so the the answer that clearly represents this number line is going to be C. So the solution to this problem is C. So I hope this video helps you in understanding how to graph and write inequality in interval notation, and I'll go ahead and see you all in the next video.

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. x/4 - 3/2 ≤ x/2 + 1

Key Concepts

Linear Inequalities

Interval Notation

Graphing on a Number Line

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