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.1/(x - 3) < 1

Accepted Answer

Solve the following rational inequality. And graph the solution set on a number line, one divided by X minus four is less than one half. So to solve this, let's first move everything to one side of the inequality. So you have one divided by X minus four is less than one half. We can move our one half over one divided by X minus four minus one half. And that should be less than zero. Now, to solve this, we need to combine these two fractions into one single fraction. Let's get the same denominator for these fractions. For our first fraction, we will multiply it by two divided by two. For our other fraction. We'll multiply by X minus four, divided by X minus four. We will get two divided by two, multiplied by X minus four minus X minus four, divided by two, multiplied by X minus four. We can combine them onto one fraction, two minus X minus four divided by two multiplied by X minus four. These are still less than zero. We can simplify it to get two minus X plus four divided by two multiplied by X minus four. And our final inequality then will be negative X plus six divided by two, multiplied by X minus four is less than zero. Now, to find our possible domains, we can set our numerator equal to zero. Our denominator equals to zero and solve. So for our numerator, we get negative X plus six equals zero, signing for X six. And we can divide it by negative one, we get X equals six can now check our denominator two multiplied by X minus four equals zero. If X minus four equals zero, adding four to both sides gives us X equals four. So now we have three intervals, we can check we have a do in from negative infinity, positive four, 46 and six to positive infinity. Now to solve this, we can use test values. Let's say we let on our first domain, X equals zero, our second domain X equals five and our final domain X equals seven. Now inequality says we want any number less than zero. So we want to find when we plug in these X values which one of these domains are negative. So far I were to plug in zero into our equation, I would get negative 3/4. If I were to plug in five, I would get positive one half. And if I were to plug in seven, I get negative 16, now there are only two domains where we have a negative answer negative infinity to four and six to infinity. Which those make up our domains or being less than zero. If we look at our possible answers, the answer to our problem turns out to be answer C which has our domain. OK. 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.1/(x - 3) < 1

Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation and Graphing Solutions

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