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In $3-14,$ solve and check each inequality.$$\frac{x}{x+5}-\frac{1}{x+5}>4$$

$\{x |-7< x <-5\}$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 8

Solving Rational Inequalities

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

06:10

In $3-14,$ solve and check…

03:36

00:57

Solve each inequality.…

05:58

01:30

Solve each equation or ine…

01:46

02:05

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00:41

01:06

we're giving this inequality X over X plus five miles, one over X plus five is greater than four, and we're going to solve it now with an inequality that has variables on the bottom. We have to start off a bit differently. That is, we're going to be multiplying both sides by the denominator to get that denominator out of here. This way of us X minus one eyes greater than four times X plus five where four times expose five. Just distribute out distributes out to four X plus 20. Okay, so now that we have this, we have to break up into two different steps. The way that these types of inequalities work where you have variables on the bottom is that you have to break up in a two different steps where one side is X being greater than zero, or rather, the denominator being greater than zero and one. And the other side is the denominator being less than zero because of the denominator were equal to zero. Then you would have a problem. You would be dividing by zero, and that's not allowed by the rules of math. So here on the left, we'll have our denominator X plus five being greater than zero. And on the right, we'll have X plus five being less than zero. Let's begin here on the left, so expose five is greater than zero. That means that we will have X is greater than negative five. And with that out of the way, let's plug in to the equation. We don't have to change anything. We just have to solve it. So X minus one is greater than four X plus 20. First off, we can subtract X from both sides. Giving us negative one is greater than three X plus 20 and then we can subtract a 20 from both sides. Giving us negative 21 is greater than three X or dividing through by. Ah, that three will give us X is less than negative. Seven. All right, so we have The X is less than negative seven. However, we already assumed that acts would be greater than negative five. And those two conditions simply can't coexist. At the same time, we can't have a number that's greater than negative five, but less than negative seven. So we know that this side of the equation just does not work or rather, this side of the process. Therefore, we can hope that we'll get an actual answer from the right side. So here we have X Plus five is Ah, less than zero, which rearranges to X is less than negative five, and we can solve the inequality by the same process as before. X minus one is, uh, except for this time we're going to be flipping the direction of our inequality symbol that is, now we have a less than instead of a greater than that's one of the most important steps to setting the denominator. Less than zero is also reversing the direction of the inequality. So we have X minus one is less than four X plus 20 and solving just the same as before, gives us negative one straighter than three x plus 20. Sorry less than I gotta remember those signs and then negative 21 is less than three. X or X is greater than negative seven, and our initial condition was that X was less than negative. Five. So we have that X is less than negative five, but greater than negative seven, and that is possible. That's very possible, like numbers such as negative six fall in that range. So we have found our answer. That is, we have the set of all X such that negative seven is less than X, which itself is less than negative five.

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