In Exercises 69–74, solve each inequality and graph the solution ...

Question

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

Accepted Answer

Hey everyone in this problem. We are given an inequality. Were asked to find X and two graph its corresponding solution set using a real number line. Ok, now the inequality we're given is two, X minus 10 divided by X plus two is less than or equal to two. Okay, let's just go ahead and write what we have two. X minus 10 Over X-plus two Is less than or equal to two. Okay. Now you're going to be inclined to multiply by X plus two. Okay, So it comes to the other side. You have to remember we're not working with an equality. We're working with an inequality. What do we have to be careful with while multiplying in an inequality? We have to be careful if we're multiplying by a negative. Okay, If we multiply by a negative, we have to flip the inequality. So what we need to do is we need to break this into two situations. Okay, let's give ourselves some more room. Now, we're going to be multiplying by X plus two. Okay. So we want to consider what happens if X plus two is positive bigger than zero, which means that X Is bigger than -2. And we also want to consider what happens if X plus two is negative less than zero. Okay, so X is less the negative two. Okay. Alright. And you'll notice I didn't use less than or equal or bigger than or equal on either of these, what happens if x is equal? So if X plus two equals zero, which means X is equal to negative two. What happens? Okay, we're all the denominator is going to be zero we're dividing by zero. So we're going to have this being an undefined situation. Okay, So x equals negative two is not going to be a solution. So we just have these two cases to consider now. Okay, so if X is bigger than negative two, X plus two is positive. So we can go ahead and multiply up and the inequality doesn't need to change. We end up with two. X minus 10 is less than or equal to two times X plus two. So we get two X minus 10. It's less than or equal to two. X plus four. Okay, we can subtract two X and we get negative 10 is less than or equal to four. Okay, well this is a true statement. Okay, and this is always true. No matter what value of X you choose in this Interval negative 10 will be less than or equal to four. Okay, so that means that any X bigger than -2 that we choose is a solution. Okay, Alright, so we have considered this first interval. We found that any X bigger than negative two is a solution. Let's move on to our second interval. Okay, so in this case we have X less than negative two. X plus two is less than zero negative. When we multiply an inequality by a negative we have to flip it. So now we get two X minus 10 is bigger than or equal to two times X plus two. Okay, we flipped that inequality because we multiplied by a negative. Now when we do this we get two X plus four on the left. Or sorry. On the right hand side. When we subtract by two X we get negative 10 is bigger than or equal to four. This is false. This is never true. No matter what x value choose negative 10 will never be bigger than four. Okay, so these are not solutions. So we have no solutions Where X is less than -2. Okay, so now let's think about what we found. We know that X equals negative two is not a solution. We know that X bigger than negative two is a solution and X less than negative two is not a solution. So let's go and draw a number line. Let's do it right here at the top. So we can still see our work. Yes, we're gonna draw a number line, we have zero here, We have negative two here. And then all of the other numbers and our solution. Let's draw it in red so we can see it. We know X bigger than negative two is a solution. X equals to a negative two. Story is not. So we're gonna have an open circle at negative two. Anything bigger. The -2 is a solution. Anything less than is not. Okay. And so our solution on a number line is going to look like this. Okay, now let's go up and compare this with the answer choices were given. And we're going to see that this matches with answer choice B. Okay. So our solution is any X value bigger than Negative Two. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

Key Concepts

Inequalities

Rational Expressions

Graphing Solution Sets

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