in-15-26-find-and-graph-the-solution-set-of-each-inequality-2x2-leq-3

Question

In $15-26,$ find and graph the solution set of each inequality.
$$
2|x+2| \leq-3
$$

Dilip Paruchuri · Accepted Answer

all right. Hey, guys. So in this problem were given two times absolute value of experts to is less than or equal to minus the reef. And so first, before we even try to simplify this inequality, let&#x27;s just take a look at it and see if it actually makes sense when you take the absolute value of anything, right, And after the value of any know. So let&#x27;s just say absolute value of some blah something right here. You&#x27;re going to get the positive value. You&#x27;re absolutely you of anything is always going to give you something greater than or equal to 00 everything abso value of zero. But any number of plug it in, it gives you something greater than zero. So what we&#x27;re saying right now is when we take our absolute value of X Plus, two were basically saying our experts to this value, whatever our absolute value of experts to is, it&#x27;s gonna be greater than zero. So we know that this is gonna be greater than zero and basically were multiplying something that&#x27;s greater than zero. Something that&#x27;s positive by another positive number two. So we&#x27;re basically essentially multiplying a positive times. A positive on the left hand side of this equation, and when you do that, you&#x27;re going to get another positive number Positive times positive is always positive. But what this inequality is saying is that it&#x27;s saying that positive number is less than or equal to minus three. This can never, ever happen. And when you&#x27;re solving inequalities like these, it&#x27;s important to check if the inequality actually makes sense that if is it, if it is even possible for the solution set to exist and one of the case, one of the best ways to check if a solution said does it does not exist is if the inequality the problem that&#x27;s given itself does not make sense, like in this case. And in this case, the problem basically breaks down to saying that a positive number is less in a negative number. So in this case, there is not going to be any solution set because no value of X doesn&#x27;t matter what value of X you choose in the real numbers. So basically saying no value of X in the reels will ever make this inequality true. And since this is the case, the solution set is empty, meaning it does not exist. There is no value of X that makes this solution true. And that is the answer to this quality problem. Thank you for listening, guys. And I hope this help. This was a new concept when you&#x27;re solving inequalities problems. I hope this helped Segway us into this new set of problems. Thank you for listening and

Dianna Johnson · Answer

Okay, so we want to solve this problem two times absolute value of X plus two is greater than Agnes three. So, in order to solve this problem, the first thing I&#x27;m gonna dio is the viable sides by two. In order to get my absolute value exposed to by itself until we get X Plus two is bigger than negative three to read it by two. Because the negative 1.5 here&#x27;s kind of the kicker about this problem. If you think about it in terms of a number line and in terms of absolute value, number is bigger than a negative. 1.5 include numbers like negative one negative, 1/2 zero, one to and three. And it keeps going on infinitely. However, this interval right here is an interval that we can&#x27;t actually achieve. We can&#x27;t show numbers like negative one or native. 1/2 night of one can&#x27;t be result of an absolute value because all absolute values yield positive numbers is their answer. And so, since this is the absolute value of exports to write, not just expose to because it was absolute value exposed to before, I actually can&#x27;t get anything in this range for my solutions and so and ordered us continue to solve this problem I have toe on. Lee use numbers in the solution set that actually are bigger than negative 1.5. So my solution shut. Technically, should be anything with values in here. Absolute values that get me a throwing bigger than negative 1.5. But when we look at that, in reality, we can&#x27;t get a negative number. So we&#x27;re actually just focusing on anything bigger than zero him. So either what&#x27;s inside this absolute value X plus two is bigger than zero or it&#x27;s smaller than zero. And why did I put that foreign? Because that&#x27;s kind of confusing. When you look at an absolute value, it could be a number that&#x27;s positive or negative on the inside of your absolute value. So, for example, negative, too absolute value of that. Be positive, too, which is still a distance greater than zero right. Positive to the absolute value of that would also be positive, too, and that also would have a distance of zero. So either whatever&#x27;s inside here is bigger than zero, or it&#x27;s a negative number, which, when you take the absolute value of it is still going to be bigger than zero. So either it&#x27;s a negative number that we get out of this or it&#x27;s a positive number. So I subtract full size by two. We have X is bigger than negative to or X is smaller than, um, subtracted from bull signs. Negative, too. So if the number is bigger than negative two or smaller than negative too, then the solutions are all numbers bigger than negative. Teoh were smaller. The negative shoes on a graph We&#x27;re just gonna draw number line here is negative. Two. It&#x27;s not equal out negative. Two. We know that because there&#x27;s no less than or equal, then sign. It&#x27;s just a plain symbol last, then looks like this alright, less that looks like that greater than looks like that less than are equal to greater than or equal to. Those are the ones that have solid lines. Since there was no line underneath, we know we&#x27;re leaving an open circle, so if we have X is greater than negative. Two things greater than negative Teoh are to the right or X is smaller than negative, too. Those things air to left, so really our answers all rial numbers except negative too

Dilip Paruchuri · Answer

all right. Hey, guys. So in this problem, we have three times absolutely of two X minus two plus two is greater than our able to minus five. So let&#x27;s solve this inequality and graphics solution. Set first things first. We want together absolute value by itself on one side of the inequality. So let&#x27;s first subtract Whoops. Let&#x27;s go with a different color pen. Let&#x27;s subtract two on both sides of the equation on the inequality on our left hand side. The two scandal out and you&#x27;re left with three times absolute value of two X minus two. And on the right hand side, you&#x27;re left with minus seven. Then you want to divide out by three on both sides because you want to get the absolute value by itself. And you wanna get rid of this three that you&#x27;re multiplying it by So on the left hand side that threes go away and you&#x27;re left with absolute value of two X minus two is greater than or equal to minus 7/3. Now let&#x27;s stop right here and take a look of what we have. We&#x27;re saying the absolute value of something is greater than or equal to minus 7/3. We already know that whenever you take the absolute value of something, it&#x27;s always gonna be greater than or equal to zero. We know that. Let&#x27;s write this in a different color pen. We already know that absolute value of two extra basically, absolutely to X minus two. We know that it&#x27;s gonna be greater than or equal to zero because you&#x27;re taking the absolute value of some random number and absolute value. Always give you something greater than or equal to zero. We also know one other thing. We know that zero is always gonna be greater than minus 7/3 because minus 7/3 is a negative number, and zero is always gonna be greater than a negative number. And if we combine these two inequalities that we have here, we can get that to X minus two. He&#x27;s gonna be greater than or equal to zero, which is greater than or equal to minus 7/3. So with without regard for what our value of X is this inequality is always true. This is always true. That absolute value of two X minus two is greater than mine is greater than or equal to minus 7/3. In a sense, a better way of saying this is absolute value of two X minus two is greater than or equal to minus 7/3 for all values of X. So basically saying any value of X, you give me just choose any value in the real plane, real numbers. This inequality is gonna be true. And that&#x27;s exactly what&#x27;s happening over here. And so if we want to write our solution set our solution said it&#x27;s gonna be all ex all X in the real numbers are basically all relax. And that and our solution said is gonna look like this. It&#x27;s gonna be the entire number line, because every value of X makes this true, and this is our answer to this problem.

Dilip Paruchuri · Answer

all right. Hey, guys. So, Ms Problem, we&#x27;re given the inequality absolute value or five have X plus two is less than zero. And right off the bat, looking at this problem, you have to You must notice that something is wrong and that something is that you&#x27;re taking the absolute value of something and you&#x27;re saying that that is less than zero. Whenever you take the actual value of any number, any variable, let&#x27;s see a variable. Why? After value, anything is always greater than or equal to zero. Always, it&#x27;s equal to zero of y zero. But any other case, absolute value of something is always greater than zero. So, in a sense, absolute value of five halves X plus two is always gonna be a positive number or zero in this case. But let&#x27;s assume it&#x27;s not zero. It&#x27;s just a positive number, and we&#x27;re saying that a positive number or a zero is basically less than zero. This is a total contradiction, does not exist, So we already know that this inequality does not make sense. And since it does not make sense, no value of axe is ever going to make this true. Basically, no X can, ah, solve this inequality. And if we really just need to graft this out, our graph of our solutions, that is just gonna be This is gonna be the real numbers. Nothing filled in. Nothing shaded in because, like I said before, no value of X is ever gonna make the absolute value of something less than zero. So thanks for listening, guys. And ah, the answer to this, by the way, just to explicitly say it out is the solution Set is empty, does not exist. Thank you for listening eyes. And I hope this helps solve this problem and understand the concept of no, um, an empty solution set when considering after value functions in.

Dilip Paruchuri · Answer

Hey, guys. So in this problem, we&#x27;re given the value the the inequality, absolute value A minus five is greater than or equal to three or after Thal for the solution set and then graft the solution set. So first thing first, let&#x27;s break up our absolute value into the two inequalities that we need to solve. So by definition, whenever you have an absolute value and inequality problem, we have to break it up into the two two conditions that we have to have in order for this inequality between met. True. What I mean by this is that absolute value a minus five greater than uncle to three is going through the same as a minus five is greater than or equal to three. And a minus five is less than are equal to minus three. These were the two inequalities that we need to solve. So a minus five greater than or equal to three, we&#x27;ll tackle the one on the left first. Now we want to isolate our eh So he has add five to both sides and we get that a is greater than or equal to eight. And on their left side we do the same thing. We add five to both sides to solve for our A and then we get that A is less than a equal to two. We can see No, sorry. A is less than or equal to To show on our graph. Let&#x27;s grab this out. So if this is our graph we have on 123456789 10 So this is our value of 2345678 This is our value of eight. And let&#x27;s make the graph our values and read so we can have the points A is equal to two and a Z equal to eight. And the first thing is saying that a is greater than or equal to eight. So anything to the right of a and including a is equal is us basically solves this inequality. And then we also have a CZ less than or equal to two. So anything to the left of to less than two and equal to two also solves this inequality. So ah, and finally weaken basically right. Our solution set as a is less than equal to two or a is greater than or equal to eight. And that graph is the graph of our solution set. So thanks for listening, guys. And I hope this helps all of this.

Dilip Paruchuri · Answer

All right. Hey, guys. So, in this problem, we&#x27;re given the inequality nine minus the absolute value. Three X plus three is less than zero. And whereas to solve for the solution set and then graft the solution set. So before we can split our inequality into two inequalities and solve for our solution set, we need to get our absolute value by itself on the left hand side of this equation. So, in order to do that, what we&#x27;re going to do is we&#x27;re going to subtract nine on both sides of the equation. Subtracting nine on the left hand side, gets rid of it, and you&#x27;re left with negative of absolute value. Let&#x27;s write that better. We&#x27;re left with negative of absolute value three X plus three less than and on the right hand side, you&#x27;re left with minus nine, and we have a negative sign attached to the absolute value that we want to get rid of. So we&#x27;re going to divide out by negative one. So on the left hand side, when you divide up by negative one, you just get absolute value of three explosive. Three on the right hand side, negative nine, divided by negative one is nine. Since you&#x27;re dividing by a negative number, the signs flipped. And so this is the absolute value side that we have absolute value inequality that we want to solve for. And we&#x27;re going to split it up into the two equations we split up into our three X plus three is greater than nine. And also three X plus three is less than negative nine. So let&#x27;s focus on the equation to the the inequality right here for us. So first, we&#x27;re going to subtract the re from both sides. And when you do that, you&#x27;re left with three X is greater than six, and then you&#x27;re gonna divide three on both sides, and that&#x27;s gonna give you X is greater than two. Now, for this equation right here, where do the same thing? We&#x27;re going to subtract three from both sides, and that&#x27;s gonna leave us with the re X is less than minus 12 and then you&#x27;re gonna divide out by Bree on both sides and that&#x27;s gonna end us up with X is less than minus fuller. And this is going to be our solution set for our inequality. And now what? We&#x27;re going to do is we&#x27;re going to graft the solution set. So let&#x27;s graft this right here. So our solutions that is going to be something like this. So let&#x27;s drip plot some lines. Hopefully we have enough lines here, So let&#x27;s say that this is minus for if this is minus four is minus three minus two minus 1012 So this is our two. And what&#x27;s gonna happen is both of these values. They&#x27;re gonna be open circles this time because we&#x27;re using greater than unless thens no equal to signs so open circles. And so let&#x27;s take a look at her first solution X is greater than two. So we&#x27;re plotting basically everything to the right of to, but not including two. And then X is less than minus four. Everything to the left of four, but not including minus four. So this is going to be our solution. Set our solutions that is gonna include X is less than minus four or X is greater than two. And this is the solution set and the graph of our solution set to this inequality. Thank you for listening. I hope this helps solve this inequality right here.

Problem

In $15-26,$ find and graph the solution set of ea…

Problem 22 Hard Difficulty

In $15-26,$ find and graph the solution set of each inequality.
$$
2|x+2| \leq-3
$$

Answer

see explanation

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Ann Xavier Gantert

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