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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q.496. (page 742)

Solve the rational inequality and write the solution in interval notation.
$\frac{2 x + 3}{x - 6} > 1$

Short Answer

Expert verified

The solution is $(- 鈭�, 9) 鈭� (6, 鈭�)$ .

Step by step solution

01

Step 1. Given information.

The given rational inequality is $\frac{2 x + 3}{x - 6} > 1$ .

02

Step 2. Simplify the inequality.

$\frac{2 x + 3}{x - 6} > 1 \frac{2 x + 3}{x - 6} - 1 > 0 \frac{2 x + 3 - x + 6}{x - 6} > 0 \frac{x + 9}{x - 6} > 0$

03

Step 4. Find all the values where the expression switches from negative to positive by setting each factor equal to zero.

$x + 9 = 0 x = - 9$ or $x - 6 = 0 x = 6$

The domain of $\frac{x + 9}{x - 6}$ is $(- 鈭�, 6) 鈭� (6, 鈭�)$ .

The required interval is:

$x < - 9$ or $x > 6$

$(- 鈭�, 9) 鈭� (6, 鈭�)$

04

Step 4. Simplified answer.

Hence, the required solution is $(- 鈭�, 9) 鈭� (6, 鈭�)$ .

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Most popular questions from this chapter

Simplify the complex rational expression by using the LCD:

$\frac{\frac{2}{x - 7} - \frac{1}{x + 7}}{\frac{6}{x + 7} - \frac{1}{x^{2} - 49}}$

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{3} + \frac{1}{x^{2}} > \frac{4}{3 x}$

Divide $\frac{x^{2} - 4 x - 12}{x^{2} + 8 x + 12} 梅 \frac{x^{2} - 36}{3 x}$

Simplify the complex rational expression by writing it as division:

$\frac{\frac{3}{4} - \frac{1}{3}}{\frac{1}{8} + \frac{5}{6}}$

Simplify the complex rational expression by using the LCD:

$\frac{\frac{1}{x^{2}} - \frac{1}{y^{2}}}{\frac{1}{x} + \frac{1}{y}}$

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