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

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

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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.

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!

Amaan M.

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John P.

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## 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!