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Chapter 2: Problem 55

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{x+1}{x+3}<2$$

Short Answer

Expert verified
The solution to the inequality \(\frac{x+1}{x+3}<2\) is \(x \in (-鈭�,-5] \cup (-3,鈭�)\).

Step by step solution

01

Simplify the inequality

To solve the inequality \(\frac{x+1}{x+3}<2\), first subtract 2 from both sides to rearrange it into a standard form. The inequality will be \(\frac{x+1}{x+3}-2<0\). Then find a common denominator and simplify the inequality. It will then become \(\frac{x+1-2(x+3)}{x+3}<0\), which simplifies to \(\frac{-x-5}{x+3}<0\).
02

Identify the critical points

The values that make the numerator zero and the denominator zero are the critical points. For this inequality, the critical points occur when \(x+5=0\) and \(x+3=0\), which are \(x=-5\) and \(x=-3\) respectively.
03

Test the intervals

To find the solution set of the inequality \(\frac{-x-5}{x+3}<0\), test each interval defined by the critical points in the simplified inequality. The intervals are \((-鈭�,-5),(-5,-3),(-3,鈭�)\). For \((-鈭�,-5)\), take \(x=-6\) as the test point. Substitution results in \(\frac{-(-6)-5}{-6+3}>0\), which is true, hence this interval is included. For \((-5,-3)\), take \(x=-4\) as the test point. Substitution results in \(\frac{-(-4)-5}{-4+3}<0\), which is false, hence this interval is excluded. For \((-3,鈭�)\), take \(x=0\) as the test point. Substitution results in \(\frac{-0-5}{0+3}>0\), which is true, hence this interval is included.
04

Express the solution in interval notation

The solution set in interval notation is \(x \in (-鈭�,-5] \cup (-3,鈭�)\). The solution implies that the values of \(x\) that satisfy the inequality are all real numbers less than or equal to -5 and all real numbers greater than -3.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)

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.

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{2 x+7}{x+3}$$

A company is planning to manufacture mountain bikes. The fixed monthly cost will be \(\$ 100,000\) and it will cost \(\$ 100\) to produce each bicycle. a. Write the cost function, \(C,\) of producing \(x\) mountain bikes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) mountain bikes. c. Find and interpret \(\bar{C}(500), \bar{C}(1000), \bar{C}(2000),\) and \(\bar{C}(4000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C} ?\) Describe what this means in practical terms.
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