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

## (x+7)/(x-1) is less than or equal to 0

# 3 Answers

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

Caity,

You have an excellent question!

The first thing I do in any fraction with variables is determine what value(s) can make the equation DNE (does not exist). In any fraction, the denominator is not allowed to equal 0. What value of x makes this denominator equal to 0? If x = 1, the denominator goes to 0, and the whole thing becomes DNE. So we know right away, that x equals 1 (x does not equal 1).

The other portion of this problem is usually handled rather easily. We have already taken care of the denominator, so all we have to do is figure out the numerator. What value for x makes the numerator less than or equal to 0?

So solve: x + 7 = 0

x + 7 - 7 = 0 - 7

x = -7.

So we handled the equals to portion. Now we need to deal with the less than portion. We can do this rather easily by testing a number larger than -7 and one smaller than -7.

Let's try -10. -10 + 7 < 0 ? YES.

Let's try -5. -5 + 7 < 0 ? NO.

So x must be less than or equal to -7, as long as x does not equal 1.

Hope this helps!

Jeremy

(x + 7)/(x - 1) ≤ 0

First, multiply both sides of the inequality by the denominator on the left-hand side to get rid of the fraction:

(x - 1)·[(x + 7)/(x - 1)] ≤ (x - 1)·0

x + 7 ≤ 0

Then, subtract 7 from both sides of the inequality to solve for x:

x + 7 - 7 ≤ 0 - 7

**x ≤ -7**

The solution in interval notation is **(-∞, -7]**, which means that the solution for x is the set of all real numbers less than -7, including -7.

# Comments

Tamara, this isn't right! You can't multiply by something in an inequality unless you know what its sign is, because you need to know whether or not you have to flip the inequality sign (that's because multiplying by a negative requires you to flip the inequality sign). When you multiplied by (x-1) on both sides and kept the inequality sign facing the same direction, you tacitly assumed that (x-1) is positive. But it doesn't have to be!

The correct answer to this question, in interval notation, is [-7,1). Read my correct explanation above, or just graph the function and look for where it lies at or below the x-axis!

## 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!

Comment