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Solve and graph the solution set.$$4 x-3>2 x+7$$...

Question

Solve and graph the solution set.$$4 x-3>2 x+7$$

Solve and graph the solution set. $$4 x-3>2 x+7$$



Answers

Solve and graph the solution set. $$4 x-3>2 x+7$$

We have to solve the inequality. X squared minus four x greater than minus tree Now we aired three boat. Besides big get X squared minus four x plus three Greater than Joe on fact rating it is X minus one X minus three. Greater than do now. Report one and three on this number line and we have to begin first region it minus infinity to the one second reading it one comma tree and tired. Reason it three to the infinity. Let's see your point V. Take a point. Deal. So van report Jiro in this inequality v get do you don't mind if one geo minus three it Quito three Richie greater than zero. So we have to take this in our solution. Now when we take the point, let's it to in this inequality vig eight to minus one. Tu minus three. It a minus one vici not greater than zero. So we have to neglect this point. No, in this interval, let's say heavy point, full vigor for my next one for my next three on following a again three, which is great reindeer. Oh, so we have toe. Consider this in other solution. So Our answer will be minus infinity to Dublin. Union three, Toe the infinity. So on this graph, this red part is ever final solution.

Hurry in orderto start isolating for X so that we can graph our solutions on a number line. Um, we can first start out by simplifying the right hand side here. I see that three that could be distributed to ah, that four minus x. So three times four would get us 12 minus three times X, which is three X and on the left hand side, we still have negative seven X plus four. All right, so from here, there's a few different ways week we could begin, um, reaching toward isolating X. So I think what I'm gonna do is add seven extra both sides because then I will eliminate any negatives. And a lot of times, that's where mistakes happen when we're solving. So on the left hand side. Now I'm left with four is greater than 12. And I have I can think of this as, like, negative three plus seven. Um, so that would get me a positive for so four x next. I want to get rid of this 12 on the right hand side. So it's a positive 12 and I need to get it down to zero. So I'm going to subtract 12. And whatever I do to one side, I need to do to the other. So I can think of this as four plus negative 12. We're four minus 12 whichever way your brain prefers, and that's gonna get me a negative eight, and that is greater than four X. So now I need to get the ex alone. So I'm goingto under the multiplication that's currently happening with that four by dividing by four. And whatever I do know, one side I need to do to the other. So negative eight divided by four will get me a negative, too. And that is greater than X because these fours canceled out. So now we Congrats on our number line. Uh, negative too is greater than X. So that means X has to be less than negative, too. So negative to will be my largest value that I need to see on the number line and because it's not included in the solution, I'm going to use parentheses on that negative too. And then I'm gonna shade everything off to the left here towards negative infinity. So when I write that an interval notation, I've got negative infinity and then negative too. And I'm using parentheses for that negative too, because it is not included in the

So to solve this absolute value inequality, the first thing we need to do is isolate the absolute value quantity itself. We knew that by subtracting two from both sides. All right, then we can write. This is a compound inequality. We know that X plus three could be less than negative five, or, um, it could be greater than five. So let's solve those two inequalities. This we're going to subtract three from both sides here. X could be less than negative. Eight. Subtract three from both sides here, X could be greater than two. So that's the solutions that right there. And we're also gonna graph that solution. Set something like that and we'll say our solutions. That starts at negative eight, but doesn't include negative A. So we use that parentheses, e there and everything to left of it. And the rest of it starts at two but doesn't include two and everything to the right

We have to find the solution set for this compound inequality and then graphic. So I have four is less than X plus four and X minus one is less than three. So I'm gonna start by subtracting four from both sides of this inequality and adding one to this inequality. Does that look of me? Zero was less than X and X is less than four. So I'm gonna write my solution set for this An interval notation. That's just gonna be 04 with parentheses on both sides because we're not including the zero or the four. And then my graph will look like this. What's your over here for here? And we're using open circles for both, not including zero or for, but just everything in between.


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