Algebra (all content)
Rational inequalities: one side is zero
Sal shows two ways to solve the inequality (x-1)/(x+2)>0. Created by Sal Khan.
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- Trying to figure out where I can find Sal giving a video on these types of equations. x+1 over 3 + x+2 over 7 = 2. What is this problem called? a linear equation or rational? Need to learn how to do these. thanks for your help(15 votes)
- Your example looks like a rational equation, Sal has 3 videos on this topic (they are called solving rational equations). I solved your equation and you have to find a common denominator first, which is 21. Then you multiply 7(x+1) + 3(x+2), all that over 21, which equals 2. After a few steps, you
ll have 10x +13=42, then 10x=29, and finally, x=10/29. I hope this helps, but like I said, Sal teaches how to solve those types of equations, just watch "solving rational equations" and youll be fine!(19 votes)
- I need help with a question such as 4/(5-3x)<0 . I can't seem to apply this knowledge to that inequality.(4 votes)
- Isolate x, so the first step would be to realize that as long as (5-3x) is a negative then 4/(5-3x) will be less than 0. So you know that 5-3x < 0 (4 can be removed). From here you can move the 3x to the other side so 5 < 3x. Divide both sides by 3 to get 5/3 < x. This can be reversed to put x on the left side and so the answer would be that x > 5/3. Vote me up if this helped :)(11 votes)
- I'm having a lot of trouble with inequalities, because I don't really understand the entire idea of an inequality... What is it? How do you solve it? The tutorial in my mathbook was almost useless... I don't really know what it is! How do you solve it? What kind of answer are you supposed to give? Is there a video where Sal explains this?(5 votes)
- Inequalities describe a relationship between two values that are not equal.
a < b states that the value of a is less than the value of b, and a > b states that the value of a is greater than the value of b.
Keep in mind that a "value" being greater or less than another value refers to its position on the number line: those with lesser values are "more negative," or further left on the number line, while those with greater values are "more positive," or further to the right on the number line.
When you are solving algebraic equations with inequalities, you treat them almost like equations. You may add or subtract on both sides without any difference.
When you multiply or divide, however, you must consider whether the operation you are performing will change the nature of the problem. Multiplying or dividing by a negative will change the signs of both sides, and thus change the relative positions of the numbers on the number line, effectively mirroring them about zero. So, if you have 2 > 1, and multiply by a -1, you get -2 > -1, which is not true, since -1 is more positive. We are forced to switch the sign, and make it: -2 < -1, in order to ensure it remains a true statement.
So -x < 1 will become x > -1, and -5x > -10 will become x < 2.
In the above inequalities, there is only one variable: x. We are only concerned with the x-values. If you choose to graph the left and right sides as separate equations, in order to find the intersection points, we are looking for when one function is above or below the other. Just keep in mind that the y-values are unimportant to the answer. The only reason we would need them is to see the relationship between the two graphs.
If you want any more practice or instruction on inequalities, here is the link to the Algebra I section: https://www.khanacademy.org/math/algebra/linear_inequalities(4 votes)
- At8:50, how can (x+2) be considered positive when -1 still falls in the range of x>-2?(3 votes)
When x=-1 x+2 = -1+2 = +1 so even though x is not positive, (x+2) is positive.
I hope that is of some help.(7 votes)
- 1/(x-2) >= 3/(x+1) i have (-2x+7)/(x-2)(x+1) but the answer is x-2/7 over denominator,,,, help(1 vote)
- 1/(x-2) >= 3/(x+1), x =/= 2, x=/= -1
1/(x-2) - 3/(x+1) >= 0
( 1(x+1) - 3(x-2) ) / ( (x-2)(x+1) ) >= 0
( x + 1 -3x + 6 ) / ( x^2 + x - 2 ) >= 0
(-2x + 7) / (x^2 + x - 2) >= 0
Since we're asked when the expression is not negative, let's make a sign chart.
Zeroes of -2x + 7:
-2x + 7 = 0
-2x = -7
x = 2/7
Zeroes of x^2 + x - 2:
Since x^2 + x - 2 was expanded from (x-2)(x+1), we can easily see that the zeroes are x = 2 and x = -1. Also remember that there are the forbidden values for x.
-2x + 7 is a line with negative slope, so on the left side of its zero it's positive and on the right side it's negative.
x^2 + x - 2 is a parabola opening up, so between its zeroes it's negative, otherwise positive.
Let's call -2x + 7 = f(x) and x^2 + x - 2 = g(x) so (-2x + 7) / (x^2 + x - 2) = f(x)/g(x) and I don't have to write that horrendously long expression.
Here's the sign chart:
f(x) + # + | - # -
g(x) + # - | - # +
--- -1 --- 2/7 --- 2 --->
f(x)/g(x) + # - | + # -
The first row is the sign of -2x + 7 in each interval, the second row is the sign of x^2 + x - 2 in each interval, and the third row is the sign of the products of the signs above, being the sign of (-2x + 7) / (x^2 + x - 2).
On the last line I marked the intervals when (-2x + 7) / (x^2 + x - 2) is not negative (that means we have to include all boundaries we can. We can't include -1 or 2 because they're forbidden but we have to include 2/7.
So the answer is x < -1 or 2/7 <= x < 2.(5 votes)
- how do you solve a linear equation using the "substitution" method?(2 votes)
- Substitution, otherwise known as "plugging in," is a method that is used in a system of equations with two or more variables. Once we know the value of one variable, we can substitute in order to find the value of another variable.
For instance, if 2x+3y=27, and we know x=6, then we would write 2(6)+3y=27, thus 3y=27-12=15, so y=5.
Substitution is also sometimes used in an equation with one variable, simply to check your answer. If we have solved 6a=18 and we found a=3, then we can write 6(3)=18.(2 votes)
- what does this sign > mean(2 votes)
- it means greater than e.g 2>1 it means 2 is bigger than 1.(2 votes)
- Hello 🤗 i need help about rational inequalities .1 over x-2 plus x over x+1 greater than equal 0 ? Could you pls help me to solve this problem ?(1 vote)
- At around00:58: Why is a>0 and b>0 the consequence of a/b>0?(1 vote)
- A/B can only be greater than 0 (in other words, positive) only if A and B are both positive (A>0 and A>0) or A and B are both negative (A<0 and B<0). Any other combination will result in a negative fraction (A/B<0 instead of A/B>0).(3 votes)
- I'm working on a problem that states: What rate of interest compounded annually is required to double an investment in 5 years. I got part of the answer as r=5th root2-1, but I have no clue how to enter this into a ti-84 to get the approximation of this 5th root. Please help soon finals!(1 vote)
- Look for a button that says x^y (that is x with the exponent of y). That is the power button. Use that and put in the reciprocal of the order of the root. For a fifth root that would be 1/5 or 0.2. A root is really a reciprocal of the power. So a square root is really the same as raising to the power of one half and a fifth root is the same as raising to the power of 1/5 or 0.2
So, if your answer is -1 + fifth root of 2, key in this:
(2 [x^y] 0.2) - 1 =
The answer is = 0.148698355... or 14.87%(2 votes)
In this video I want to do a couple of inequality problems that are deceptively tricky. And you might be saying, hey, aren't all inequality problems deceptively tricky? And on some level you're probably right. But let's start with the first problem. We have x minus 1 over x plus 2 is greater than 0. And I'm actually going to show you two ways to do this. The first way is, I think, on some level, the simpler way. But I'll show you both methods and whatever works for you, well, it works for you. So the first way you can think about this, if I have just any number divided by any other number and I say that they're going to be greater than 0. Well, we just have to remember our properties of multiplying and dividing negative numbers. In what situation is this fraction going to be greater than 0? Well, this is going to be greater than 0 only if both a and-- so we could write both a is greater than 0 and b is greater than 0. So this is one circumstance where this'll definitely be true. We have a positive divided by a positive; it'll definitely be a positive. It'll definitely be greater than 0. Or we could have the situation where we have a negative divided by a negative. If we have the same sign divided by the same sign we're also going to be positive. So or-- a is less than 0 and b is less than 0. So whenever you have any type of rational expression like this being greater than 0, there's two situations in which it will be true. The numerator and the denominator are both greater than 0, or they're both less than 0. So let's remember that and actually do this problem. So there's two situations to solve this problem. The first is where both of them are greater than 0. If that and that are both greater than 0, we're cool. So we could say our first solution-- maybe I'll draw a little tree like that-- is x minus 1 greater than 0 and x plus 2 greater than 0. That's equivalent to this. The top and the bottom-- if they're both greater than 0 then when you divide them you're going to get something greater than 0. The other option-- we just saw that-- is if both of them were less than 0. So the other option is x minus 1 less than 0 and x plus 2 less than 0. If both of these are less than 0 then you have a negative divided by a negative, which will be positive. Which will be greater than 0. So let's actually solve in both of these circumstances. So x minus 1 greater than 0. If we add 1 to both sides of that we get x is greater than 1. And if we do x plus 2 greater than 0, if we subtract 2 from both sides of that equation-- remember I'm doing this right now-- we get x is greater than minus 2. So for both of these to hold true-- so in this little brown or red color, whatever you want to think of it-- in order for both of these to go hold true, x has to be greater than 1 and x has to be greater than minus 2. This statement we figured out means that x has to be greater than 1 and this statement tells us that x has to be greater than minus 2. Now, if x is greater than 1 and x has to be greater than minus 2, x clearly has to be greater than 1. You know, 0 would not satisfy this because 0 is greater than minus 2, but it's not greater than 1. So for something to be greater than 1 and minus 2 it has to be greater than 1. This whole chain of thought where I'm saying the numerator and the denominator are greater than 0, that's only going to happen if x is greater than 1. Because if x is greater than 1, then x is definitely going to be greater than negative 2. Any number greater than 1 is definitely greater than negative 2. So that's one situation in which this equation holds true, and we can even try it out. Let's say x was 2. 2 minus 1 is 1 over 2 plus 2. It's 1/4. It's a positive number. Now let's do the situation where both of these are negative. If the x minus 1 is less than 0, if we add 1 to both sides of that equation that tells us so x minus 1 is less than 0. That's the same thing-- if we add 1 to both sides of that-- as saying that x is less than 1. So that constraint boils down to that constraint. Now this constraint, x plus 2 is less than 0. If we subtract 2 from both sides we get x is less than minus 2. So this constraint boils down to that constraint. So in order for both of these guys to be negative, both the numerator and the denominator to be negative-- we know that x has to be less than 1 and x has to be less than minus 2. Now if something has to be less than 1 and it has to be less than minus 2, well, it just has to be less than minus 2. Anything less than minus 2 is going to satisfy both of these. So this boils down to just x could also be less than minus 2. And remember, this is an or. Either both the numerator and the denominator are positive, or they're both negative. So both of them being positive boil down to x could be greater than 1, or both of them being negative boils down to x is less than minus 2. So our solution is x could be greater than 1 or-- that's both of these positive-- or x is less than minus 2. That's both of these negative. And if we wanted to draw it on a number line-- let me draw a number line just like that. That could be 0 and then we have 1, so x could be greater than 1. Not greater than or equal to. So we put a little circle there to show that we're not including 1. And everything greater than 1 will satisfy this equation. Or anything less than minus 2. So we have minus 1, minus 2, anything less than minus 2 will also satisfy this equation by both making the numerator and denominator negative. And you could try it out. Minus 3. Minus 3 minus 1. I'll just do minus 3 minus 1 is equal to minus 4. And then minus 3 plus 2. Minus 3 plus 2 is equal to minus 1. Minus 4 divided by minus 1 is positive 4. So all of these negative numbers also work. Now, I told you that I would show you two ways of doing this problem. So let me show you another way if you found this one maybe a little bit confusing. So the other way-- let me rewrite the problem. You get x minus 1 over x plus 2 is greater than 0. And actually, let's mix it up a little bit and you could apply the same logic. Let's say it's greater than or equal-- well, actually no. I'll just keep it the same way and maybe in the next video I'll do the case where it's greater than or equal just because I really don't want to-- maybe I want to incrementally step up the level of difficulty. We're just saying x minus 1 over x plus 2 is just straight up greater than 0. Now one thing you might say is well, if I have a rational expression like this, maybe I multiply both sides of this equation by x plus 2. So I get rid of it in the denominator and I can multiply it times 0 and get it out of the way. But the problem is when you multiply both sides of an inequality by a number-- if you're multiplying by a positive you can keep the inequality the same. But if you're multiplying by a negative you have to switch the inequality, and we don't know whether x plus 2 is positive or negative. So let's do both situations. Let's do one situation where x plus 2-- let me write it this way. x plus 2 is greater than 0. And then another situation where-- let me do that in a different color. Where x plus 2 is less than 0. These are the two possibilities for x plus 2. Actually, in those situations can x plus 2 equal 0? If x plus 2 were to be equal to 0 than this whole expression would be undefined. And so that definitely won't be a situation that we want to deal with it. It would an undefined situation. So these are our two situations when we're multiplying both sides. So if x plus 2 is greater than 0 that means that x is greater than minus 2. We can just subtract 2 from both sides of this equation. So if x is greater than minus 2, then x plus 2 is greater than 0. And then we can multiply both sides of this equation times x plus 2. So you have x minus 1 over x plus 2 greater than 0. I'm going to multiply both sides by x plus 2, which I'm assuming is positive because x is greater than minus 2. Multiply both sides by x plus 2. These cancel out. 0 times x plus 2 is it just 0. You're just left with x minus 1 is greater than-- this just simplified to 0. Solve for x, add 1 to both sides, you get x is greater than 1. So we saw that if x plus 2 is greater than 0, or we could say, if x is greater than minus 2, then x also has to be greater than 1. Or you could say if x is great-- well, you could go both ways in that. But we say, look, both of these things have to be true. If for x to satisfy both of these, it just has to be greater than 1. Because if it's greater than 1 it's definitely going to satisfy this constraint over here. So for this branch we come up with the solution x is greater than 1. So this is one situation where x plus 2 is greater than 0. The other situation is x plus 2 being less than 0. If x plus 2 is less than 0 that's equivalent to saying that x is less than minus 2. You just subtract 2 from both sides. Now, if x plus 2 is less than 0, what we'll have to do when we multiply both sides-- let's do that. We have x minus 1 over x plus 2. We have some inequality and then we have a 0. Now if we multiply both sides by x plus 2, x plus 2 is a negative number. When you multiply both sides of an equation by a negative number you have to swap the inequality. So this greater than sign will become a less than sign because we're assuming that the x plus 2 is negative. These cancel out. 0 times anything is 0. We get that x minus 1 is less than 0. Solving for x, adding 1 to both sides, x is less than 1. So in the case that x plus 2 is less than 0 or x is less than minus 2, x must be less than 1. Well, I mean we know-- if you say something has to be less than minus 2 and less than 1, just saying that it's less than minus 2 will do the job. Anything less than minus 2 is going to satisfy this one. But not anything that satisfies this one is going to satisfy that one. So this is the only constraint we have to worry about. So in the event where x plus 2 is less than 0, we can just say that x has to be less than minus 2. That'll satisfy this equation. So our final result is x is either going to be greater than 1 or x is going to be less than minus 2. So once again, I can graph it on the number line. x is greater than 1 right there. You have 0, minus 1, minus 2. And then you have x is less than minus 2, you're not including the minus 2. And just like that. And that is the exact same result we got up here. So whichever version you find to be easier. But you can see, they're both a little bit nuanced and you have to think a little bit about what happens when you multiply or divide by positive or negative numbers.