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 - 2)/(x + 2) ≤ 2

Accepted Answer

All the following rational inequality. And graph the solution set on the number line where we have x minus four over X plus four is less standard equal to five. Now we want to go ahead and simplify this equation a little bit. We want this to equal zero for instance to consult for X. Let's go ahead and move our five were first. So subtract five from both sides. You get x minus four X plus four minus five. Now I want to combine my five and my fraction to do that. We're gonna turn our five into a fraction. By multiplying by the denominator it's minus four, X plus four minus five times X plus four divided by X plus four Less than equal to zero. This will allow us to combine our fractions. You'll get X -4 -5 times x plus four over X plus four is less than equal to zero. Now we can go ahead and simplify some more. X minus four minus five. X minus 24. X plus four equals zero. Finally negative four X minus 24 over X post four is less than equal to zero. Now we want to go ahead and check our intervals in zeros. Let's go ahead and find the zeros of this. Let's go and set the denominator equal to zero first. So you get four x minus 24. The four X is equal to zero Solomon for X. four, x equals 24 or x equals negative six. We'll do the same for the denominator. X plus four is equal to zero, X is equal to negative four. So interval in check. So -6. Negative four. We can also check to other intervals. Everything above negative four and everything below negative six. We're gonna choose a number and each one of these intervals to test to see what's negative because we're looking for less than zero in this case. Let's go ahead and make a little test chart here. Play test X and sign Our interval firstly we can use infinity to negative six. Let's say we use F of negative seven first. If we were to plug that into our equation we get the answer negative four or three that is negative. Our next interval there's some native six to negative four Which is a number on that interval. We'll choose negative five, Plug that in. We get positive four. So we get a positive. Finally we'll check our final interval negative 4 to infinity. Let's say f is equal to zero. X is equal to zero. Rather Plug that in. We get -6. That's also minus. Now we want everything that is less than zero. So everything that is negative, which means the intervals we want Are negative infinity to negative six and -4 to Infinity. And we will include our native six as possible solution which means to use a square bracket. If you look at our possible answers this means the answer to our problem turns out to be answer a negative six and negative four to infinity. Okay, I hope to help you solve the problem. Thank you for watching. Goodbye.

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 - 2)/(x + 2) ≤ 2

Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation and Graphing Solutions

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