Saw that- is if both of them were less than 0. You're going to get something greater than 0. They're both greater than 0 then when you divide them Minus 1 greater than 0 and x plus 2 greater than 0. Solution- maybe I'll draw a little tree like that- is x So let's remember that andĪctually do this problem. There's two situations in which it will be true. Of rational expression like this being greater than 0, So or- a is less than 0Īnd b is less than 0. If we have the same signĭivided by the same sign we're also going to be positive. Where we have a negative divided by a negative. We have a positive dividedīy a positive it'll definitely be a positive. Is greater than 0 and b is greater than 0. Greater than 0 only if both a and- so we could write both a This fraction going to be greater than 0? Well, this is going to be Our properties of multiplying and dividing negative numbers. Number and I say that they're going to be greater than 0. So the first way you can thinkĪbout this, if I have just any number divided by any other But I'll show you both methodsĪnd whatever works for you, well, it works for you. Hey, aren't all inequality problems deceptively tricky? And on some level We can't include -1 or 2 because they're forbidden but we have to include 2/7.Ĭouple of inequality problems that are deceptively tricky. 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. 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). 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. X^2 + x - 2 is a parabola opening up, so between its zeroes it's negative, otherwise positive. 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. Also remember that there are the forbidden values for x. 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. Since we're asked when the expression is not negative, let's make a sign chart. If you want any more practice or instruction on inequalities, here is the link to the Algebra I section: The only reason we would need them is to see the relationship between the two graphs. Just keep in mind that the y-values are unimportant to the answer. 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. In the above inequalities, there is only one variable: x. We are forced to switch the sign, and make it: -2 -1, and -5x > -10 will become x < 2. So, if you have 2 > 1, and multiply by a -1, you get -2 > -1, which is not true, since -1 is more positive. 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. When you multiply or divide, however, you must consider whether the operation you are performing will change the nature of the problem. You may add or subtract on both sides without any difference. When you are solving algebraic equations with inequalities, you treat them almost like equations. 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. Inequalities describe a relationship between two values that are not equal.Ī b states that the value of a is greater than the value of b.
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