Solve each rational inequality in Exercises 43–60 and graph th...

Question

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x−3)>0

Accepted Answer

Welcome back. I am so glad you're here. We are Given the rational inequality, X divided by x minus eight is greater than zero. So the first step we want to have for solving a rational inequality is to have one side equal to zero, which it already is, which is great. So we don't have to do Any rearranging there. The next step is that we want to set both of our numerator and our denominator equal to zero. So we'll have X. That's the numerator equal to zero and X minus eight equals to zero. In doing this. We're going to find our boundaries for our intervals. So we'll solve for X for both of these, the first ones already done, we have X equals zero for one boundary and then we'll add eight to both sides for this X minus eight equals zero and we'll get X equals eight. So that means that on a number line We've got boundaries at zero and eight giving us the intervals from negative infinity. All the way to zero from 0 to 8 and from eight to positive infinity. And what we want to do next is choose a test point from each of those three intervals. So for a test point for less than zero, I'm going to choose negative one. For a test point between zero and eight, I'm going to choose one. I'm not putting it where it would go in the number line and for a test point that is larger than eight, I'm going to choose nine. And now what we're going to do is plug in those test point values in for X into our original inequality And we're going to see where it is true that X divided by x minus eight is greater than zero. So let's start off with negative one instead of X. In the numerator it will be a negative one in the numerator over. Not X minus eight, but negative one minus eight is greater than zero. So our numerator is negative one. The denominator negative one minus eight is negative nine and negative one over negative nine is going to leave us with a positive 1/9. Is that greater than zero? Yes, 1/9 is greater than zero. So this section is true for numbers that are less than zero. Our inequality is true. Now let's try positive one. So instead of X over x minus eight it's 1/1 minus eight. And simplifying the denominator one minus eight is negative seven. So we have a negative one. Seventh is greater than zero. Is that true? Is negative 1/7 greater than zero, nope not true. So the inequality is not true for numbers between zero and eight. Now let's check Nine. We've got 9/9 minus eight. Simplifying our denominator, we've got nine minus eight is one. So we've got nine is greater than zero. Is that true? Yes it is nine is greater than zero. So for numbers greater than eight all the way up to infinity. This inequality is true. What about at zero and at eight? Well, we know that 40 that that is going to make the inequality equal to zero. So we take a look at our inequality sign and its strict. It's saying it doesn't include ones that are equal to zero only for numbers that are greater than zero. So for zero we are going to have a parentheses to indicate that we are not including zero. But then we draw a line going down toward negative infinity because it does include all the numbers less than zero. Then we skip the numbers from 0 to 8. Because those make it untrue now, how about eight? What's happening at eight itself? While we know that we got eight? When we set the denominator equal to zero and when the denominators equal to zero, the fraction is undefined. So it cannot be eight. That would make this whole thing undefined. So again, we're going to put a parentheses to indicate No, it cannot be eight. You are not included eight. But then we will draw a line going all the way up to positive infinity. While an arrow pointing toward positive infinity saying that for numbers greater than eight, they are true for this inequality. Now, how do we say this? In interval notation we can put a parentheses negative infinity comma zero. And then parentheses to say not including zero. In union with parentheses positive eight parentheses says not including eight. All the way up to positive infinity parentheses. We look at our answer choices and this matches with answer choice a well done, we'll catch you on the next one.

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x−3)>0

Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation and Graphing Solutions

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