solve the rational inequality. express the solution using interval notation. (x+7)/(x-1) is less than or equal to 0

Hi Caity,

Unfortunately neither Tamara nor Jeremy gave you a correct answer, much less a correct explanation. There are two ways you can solve this inequality:

1) You can interpret the problem as asking when f(x) is less than or equal to 0, where f(x) is the function f(x)=(x+7)/(x-1). So one way to solve the problem would just be to graph the function f(x)=(x+7)/(x-1) (either by hand, if you know how, or on your graphing calculator, if you don't) and look for the x values where the graph is at or below the x axis (because you wanted the function to be less than or equal to zero, which corresponds to y values that are less than or equal to zero, which corresponds to being at or below the x axis). If you do this, you'll see the function only dips below the x axis between -7 and 1. So the correct answer, in interval notation, is [-7,1). In inequality notation, this says -7<=x<1. You get to include -7 in the answer because the inequality said f(x) was allowed to be 0.

2) You can also solve this problem by hand, without graphing. The step-by-step procedure is always the same:

**Determine the roots of the numerator and the denominator.**In your case, the numerator has one root, at x=-7, and the denominator has one root, at x=1.**Draw a number line and plot the roots you just found**. By "number line" I just mean a horizontal line, and by "plot" the roots I just mean mark off the roots you just found. In this case you should just draw -7 and 1.**Notice that the two roots break up the number line into three different regions: the region to the left of -7, in between -7 and 1, and to the right of 1.****Choose a number in each of the resulting regions and plug those numbers into the function to test whether the inequality is true in that region**. In our example, you could choose the number -8 for the first region (to the left of -7), 0 for the second region (in between -7 and 1), and 2 for the third region (to the right of 1). If you plug -8 into the function, you get 1/9, which is NOT less than or equal to 0,*so that region doesn't work*. For the second region, if you plug in the number we chose, namely 0, you get -7, which IS less than or equal to 0,*so that region works*. Finally, if you plug in the number we choose for the third region, namely 2, you get 9, which is NOT less than or equal to 0,*so that region doesn't work*.**The solution set to the inequality is just the set of all the regions that work**. In our case, the only region that worked was in between -7 and 1. We have to be careful about whether to include the endpoints, i.e. whether to include -7 and 1. We should include -7 because the inequality allowed the function to equal zero, but we should NOT include 1 because the function isn't defined there. So the final answer is -7<=x<1, or in interval notation [-7,1).

This is a general procedure: you can use it to solve any inequality involving a rational function. Of course, all I've done is tell you what steps to follow in order to get the solution -- I haven't told you WHY those steps actually give you the right answer.
If you're curious, please leave a comment and I'll explain more. It has something to do with an important theorem called the
**Intermediate Value Theorem**!

Best wishes,

Matt L.

B.S., Theoretical Mathematics, MIT

## Comments

Jeremy, this isn't right! The correct answer is -7<=x<1, or, in interval notation, [-7,1). You've mistakenly only analyzed the numerator of the function, but you have to analyze the entire function!