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

Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
5/(1-x)≤2/(1-x)

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. The inequality is 10 divided by five minus X which is less than or equal to three divided by five minus X. And our answer choices are A from negative infinity to five B from five to infinity. See from negative infinity to negative five in union with five to infinity and D from negative 5 to 5. Alright. So the first step we recall from previous lessons in solving rational inequalities is to set one side equal to zero. So let's subtract 33 divided by five minus X from both sides of the equation. So -3/5 - and that way the right hand side will be equal to zero. So on the left, we'll have 10 divided by five minus X minus three, divided by five minus X is less than or equal to. And then on the right hand side, three divided by five minus X minus three divided by five minus X equals zero. Great. The first step is done. The next step is that we want the side with our fractions to have a single fraction. So that means we need to subtract our two fractions to get them combined to be one fraction. In order to subtract fractions, we need a common denominator. And yay, we already have a common denominator. So we can go ahead and subtract our numerator 10 minus three equals seven And then our denominator remains 5 -1 and that's still less than or equal to zero. Okay, first step is done. The second step is we ask ourselves what values would cause our numerator and or the denominator to equal zero. Well, the only part of our fraction that has a variable in it is the denominator. So we're going to set the denominator equal to 05 minus X equals zero. Now we're going to solve for X. So we can subtract five from both sides of the equation. And then we'll have negative X equals negative five. And now we can divide both sides by negative one. And on the left, negative X divided by negative one equals a positive X and on the right, negative five divided by negative one equals a positive five. So our X value is 55 is what would make our denominator equal to zero. And therefore five is the point for our intervals. It creates our intervals for us. And I'll show you what that means. We draw a number line from negative infinity to positive infinity and five is going to create an interval for us, we have one interval from negative infinity to five and another interval from five to infinity. So now we want to see which one is true for our original inequality. In order to see which one is true, we choose a test point from each of the two intervals from negative infinity to five. I am going to choose the test 50.0 and from five to positive infinity, I am going to choose the test 50.6 and now we fill those in to the original inequality. So we have 10 divided by five minus no longer X but now zero for the first test point is less than or equal to three divided by five minus no longer X. But now zero, simplifying our denominators five minus zero is five for both of them. So we have 10/5 is less than or equal to 3/5. Simplifying on the left 10/5 is two is less than or equal to 3/5. Is it true that two is less than or equal to 3/5? Not at all two is greater than 3/5. So the test 20.0 doesn't work, meaning that the interval from negative infinity to five is not true for our inequality. We can throw six in for X to ensure that that does work. So we'd have 10/5 minus six is less than or equal to 3/5 minus. Again six and five minus six in the denominator is going to be negative one. So we have 10 over negative one is less than or equal to three over negative one and anything divided by negative one is going to be itself. But flip the sign so we have 10 divided by negative one is negative 10 is less than or equal to three divided by negative one is negative three. So is it true that negative 10 is less than or equal to negative three? Absolutely. It is. So this test point shows that it is true from five to infinity for our inequality. Now, how do we express that in interval notation? Well, we've got a five comma infinity. But are we talking about parentheses or brackets? Well, infinity is always going to be closed with a parentheses because we'll never quite obtain infinity. But for five, is that a parentheses or brackets, we typically look at the original inequality and we see here that we've got less than or equal to. However, we're dealing with rational inequalities. And there's another step we need to ask ourselves where that five came from. And the five came from when we were looking at the denominator and saying, well, what would make this denominator equal to zero? And if we plug a five in there for X in the denominator five minus five that equals zero. And for a fraction, if we have zero, we would have an undefined fraction. So it cannot be five, X cannot equal five. So to say that five is not included, we put a parentheses instead of a bracket. Alright. So we're looking for from five to infinity, we look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 5/(1-x)≤2/(1-x)

Key Concepts

Rational Inequalities

Interval Notation

Critical Points

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