/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 Solve each inequality in Exercis... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 68

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

Short Answer

Expert verified
The solution set of the inequality \(\frac{1}{x+1}>\frac{2}{x-1}\) is \(( -鈭�, -3)\)

Step by step solution

01

Clear the Fractions

Multiply each side of the inequality by \((x+1)(x-1)\) to clear the fractions. \((x+1)(x-1)\times \frac{1}{x+1} > \((x+1)(x-1)\times \frac{2}{x-1}, resulting in \((x - 1) > 2(x + 1)\).
02

Simplify the Inequality

Distribute the 2 on the right hand side to get \(x - 1 > 2x + 2.\) Move all terms to one side to form a quadratic inequality: \(2x + 3 < x\).
03

Solve the Quadratic Inequality

Subtract x from both sides to isolate the variable on one side: \( x + 3 < 0.\) Solution to the inequality is \( x < -3\).
04

Test Intervals around the Critical Point

Choose test points from each of the intervals divided by the critical point -3. For x < -3, choose, for instance, x = -4. Substituting x = -4 into the original inequality gives a true statement. Thus, the interval \(( -鈭�, -3)\) satisfies the inequality.
05

Graph the Solution Set

On the number line, an open circle (indicating 'less than') is plotted at -3 and an arrow points towards -鈭�. This represents the solution set for the inequality.

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Most popular questions from this chapter

Use a graphing utility to graph \(y=\frac{1}{x^{\prime}}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function's graph.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}\)

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3} $$

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Divide 737 by 21 without using a calculator. Write the answer as quotient \(+\frac{\text { remainder }}{\text { divisor }}\)
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