Solve each rational inequality. Give the solution set in inter...

Question

Solve each rational inequality. Give the solution set in interval notation. 5/(x+1)>12/(x+1)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to solve the following inequality and express the answer in interval notation. Our inequality is two divided by X plus seven is greater than nine divided by X plus seven. And our answer choices are from negative infinity to negative seven. That's answer choice. A an answer choice B is from negative seven to infinity. Answer choice C is from negative infinity to negative seven in union with seven to infinity. And answer choice D is from negative seven to positive seven. Alright. Well, we recall from previous lessons that in order to solve rational inequalities. The first step is we want one side set equal to zero. So I am going to subtract two over X plus seven from both sides of the equation. So now on the left hand side two over X plus seven minus two, over X plus seven equals zero. And that is still greater than, and on the right hand side, we have nine over X plus seven minus two over X plus seven. And now still part of this first step, we want to ensure that the side that is not zero has only one fraction. So we need to subtract these two fractions on the right hand side. And in order to subtract two fractions, they need to have a common denominator. And in this case, they do X plus seven is a common denominator. Yay Snow, we can go ahead and subtract our enumerators nine minus two equals seven. So we have seven over X plus seven and that right hand side is less than zero. Alright. The next step is we ask ourselves what values would cause either the numerator and or the denominator to equal zero. And here we only have a variable in the denominator. So we're going to set the denominator X plus seven equal to zero, solve for X and find out what values would make our denominator equal zero. While for X plus seven equals zero, I'm going to subtract seven from both sides of the equation. And then on the left we have X equals and on the right, we have negative seven. What does that mean? Well, now negative seven becomes our boundary. And so if we draw a number line, we have from negative infinity to positive infinity and we have a boundary at negative seven and we have now created two intervals, one from negative infinity to negative seven and the other from negative seven to positive infinity. Now we need to choose a test point from each of the two intervals. So for from negative infinity to negative seven I'm going to choose the test point negative eight and from the interval negative seven to positive infinity, I'm going to choose the test 70.0. Now we're going to plug the two test points in for X in our original inequality and see which test point is true. So plugging negative eight and we have to over not X but negative eight plus seven is still greater than nine over not X but negative eight plus seven. And simplifying our denominators, negative eight plus seven is negative one. So we have two over negative one is greater than nine, over negative one and two divided by negative one is negative two, nine divided by negative one is negative nine. Is it true that negative two is greater than negative nine? Yes, it is. So that means our test point told us that the values from negative infinity to negative seven are true for this inequality. Now let's test zero. We have two divided by not X but zero plus seven is greater than nine divided by not X but zero plus seven. Simplifying our denominators zero plus seven is seven. So we have to over seven is greater than 9/7. Is it true that 2/7 is greater than 9/7? Nope, not true. So that tells us our test point tells us that for values from negative seven to positive infinity, our inequality is not true. So we're left with negative infinity to negative seven. How do we express that in interval notation? Well, we can put negative infinity comma negative seven. But are we talking about parentheses or brackets? Well, negative infinity will always be closed with parentheses because we'll never quite obtain negative infinity. How about negative seven? Well, where did our negative seven come from? It came from the denominator. If we set X equal to negative seven and plug that in negative seven plus seven, our denominator would be equal to zero. And if a denominator equal zero, it's undefined. We can't have that. So negative seven, you are not invited, you get closed with a parentheses as well to say that you are not included. Now, we look at our answer choices for from negative infinity to negative seven and that matches with answer choice a well done. We'll catch you on the next one.

Solve each rational inequality. Give the solution set in interval notation. 5/(x+1)>12/(x+1)

Key Concepts

Rational Inequalities

Domain Restrictions

Interval Notation and Solution Sets

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