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In $3-14,$ solve and check each inequality.$$3-\frac{2}{a+1}<5$$

$\{a | a<-2 \text { or } a>-1\}$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 8

Solving Rational Inequalities

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Campbell University

Oregon State University

McMaster University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

05:58

In $3-14,$ solve and check…

08:57

03:36

03:13

01:18

02:05

03:42

02:19

08:23

02:28

Solve each equation or ine…

in this problem, writers were given the inequality three minus two over equals one is less than five and were asked to solve the inequality and give a solution set for the expert variable A. That makes the inequality true. And also to verify the solutions that as well. So let's start by bringing everything over to one side of the equation. Since we have our variable in the denominator, we're gonna have to do an extra step in the process like the previous problem. So what we're gonna do first use, we're gonna combine everything on left side like I said. So first step is gonna be to bring the night the five over in the left hand side. We do that by subtracting five on both sides. Subtracting pile on the left hand side just gives us zero on the right hand side. We're left with minus two minus two over. We want to combine everything into a woman rat of one left. Remember here. So what we're going to do is we're gonna buy the common denominator of a plus. What? So when you do that, you get minus to a minus two over eight plus moderne minus two over a plus one. Just a rewrite with re written form of this stuff right here. Now we combine it and we get minus to a minus for over a plus One is less than zero. I'm gonna bring my work up to hear from more space. Next up we can do is we can factor out of minus two just to make our life a little bit easier. Uh, it's nicer to work with cleaner functions. So you get minus two times a plus two over a plus one is equal to up, not equal to is less than zero. And then if you multiply by negative 1/2 on both sides So let's multiply by negative. So let's do this differently. So what they were It's gonna look kind of weird kind of crammed. So what I'm gonna do is I'm gonna write. This is a secret second step. So if we do negative 1/2 times negative to over times a plus two over a plus one less than zero times negative one have we get rid of the negative too in this multiplication stuff and you're you're left with a plus two over a plus, with one is greater than zero now because we must live by a negative number. Like the previous problem, we're gonna set our new major denominator equal to zero, and we're gonna solve for the other coins. And then we're just gonna plug in values from each region to check whether the inequality holds. So let's go on to a separate board right here. Did you find our pivot points? So are better points we're gonna be. So this is our inequality. What are Inequality has become to a plus. Two is greater than zero. So we're going to solve the equations. A plus two equals zero and a close one equals zero. This'd zero. And the values that we get is a who's minus to an a equal minus one. So on our number likes to run till didn't work. So we have three regions one, two on three on. Let's pick a value from each region, see which one works. So in Region one, let's choose negative three. So if you choose 93 it this is gonna become night of three plus two over negative three plus one. Uh, this is gonna simplify to let's write that again at one does not look it. So this is gonna be called negative one on the numerator. Negative two on the denominator, which is 1/2. And that's greater than zero. No, we know that this works. Region one is a check that is a part of the solutions. Now let's take a look at region too. And in Region two, we're gonna use the value negative through. Perhaps so when we do negative three halves plus two over negative three halves plus one, we get minus 1/2. I mean, uh, apologies. 1/2 in the numerator and minus 1/2 of the denominator that simplifies the negative one. And that if we set the inequality, it's supposed to be graded on zero. We know that's not true. So we know that region to does not work now. Finally, if we take a look at Region three, Region three, let's choose value zero. When we plug in zero, we played in zero plus two over zero plus one. That gives us two over one, which gives us two, which is greater than zero. So that is a part of the solution. Is that and solutions that three works. So let's draw this on the ground. We use open circles here because this is a greater than sign not a greater than or equal to on our solution set on a graph graft. Looks like this. All values to the left of minus two. All values to the right of minus one. So if we want to write out our solution gonna look like this, it's gonna be a less than minus two or a greater than minus one. This is the answer to the inequality problem. And this all this step right here is our verification process. Thanks for listening, guys. And I hope this helps solve this inequality.

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