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Chapter 2: Problem 56

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{x}{x-1}>2$$

Short Answer

Expert verified
According to the steps provided above, the solution to this exercise is \(x \in (-\infty,1)\), excluding 1.

Step by step solution

01

Subtract 2 from both sides

The inequality can be turned into a simpler form by subtracting 2 from both sides, which results in: \(\frac{x}{x-1} - 2 > 0\). So, manipulate it into a single fraction form by finding a common denominator, here it is (x-1).
02

Simplify the inequality

Once we have a common denominator, continue to simplify the inequality which is done by subtracting throughout. This results in finding a new inequality \(\frac{x-2(x-1)}{x-1} > 0\) This simplifies to \(\frac{x-2x+2}{x-1} > 0\), so our new inequality becomes: \(\frac{-x+2}{x-1} > 0\).
03

Choose test points and create intervals

To solve the inequality, one needs to find the critical values and then take test points in the subintervals. In this case, critical values are the roots of the numerator and denominator function. Here, values are 1 (denominator) and 2 (numerator). Divide the number line into intervals based on these critical numbers, so intervals are \(-\infty,1\), \(1,2\) and \(2,\infty\). Select test points from each interval (like 0, 1.5 and 3) and substitute those values into the simplified inequality.
04

Solve the inequality based on the sign of each interval

Substituting the test points in the simplified inequality \(\frac{-x+2}{x-1}\): for x=0 we get 2 which is greater than 0, so the interval \(-\infty,1\) is part of the solution. For x=1.5 we get -1 which is less than 0, hence the interval \(1,2\) is not part of the solution. For x=3, we get -1/2 which is less than 0 so the interval \(2,\infty\) is not part of the solution either.
05

Graph the solution and express it in interval notation

The solution will be shown on a number line with an open circle at x = 1 (unincluded in the solution) and a shading to the left to indicate the solution lies in that region. The solution in interval notation, given these findings, will be \(-\infty,1\)

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Most popular questions from this chapter

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000\). It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C,\) of producing \(x\) pairs of shoes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), \bar{C}(10,000),\) and \(\bar{C}(100,000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C} ?\) Describe what this represents for the company.

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