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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.478. (page 741)

Solve the rational inequality and write the solution in interval notation.
$\frac{4}{x - 2} < \frac{3}{x + 1}$

Short Answer

Expert verified

The solution is $x < - 10 o r - 1 < x < 2$ .

Interval notation is $(- 鈭�, 10) 鈭� (- 1, 2)$ .

Step by step solution

01

Step 1. Given information.

The given inequality is $\frac{4}{x - 2} < \frac{3}{x + 1}$ .

02

Step 2. Simplify the inequality.

$\frac{4}{x - 2} < \frac{3}{x + 1} \frac{4}{x - 2} - \frac{3}{x + 1} < 0 \frac{4 x + 4 - 3 x + 6}{(x + 1) (x - 2)} < 0 \frac{x + 10}{(x + 1) (x - 2)} < 0$

03

Step 3. Find all the values where the expression switches from negative to positive by setting each factor equal to zero.

$\begin{array}{rcl} x + 10 & = & 0 \\ x & = & - 10 \end{array}$ or $\begin{array}{rcl} x + 1 & = & 0 \\ x & = & - 1 \end{array}$ or $\begin{array}{rcl} x - 2 & = & 0 \\ x & = & 2 \end{array}$

The domain of $\frac{x + 10}{(x + 1) (x - 2)}$ is $(- 鈭�, - 1) 鈭� (- 1, 2) 鈭� (2, 鈭�)$ .

04

Step 4. Simplified answer.

The solution consists of all of the true intervals.

$x < - 10 o r - 1 < x < 2$

Interval notation:role="math" localid="1645082052915" $(- 鈭�, 10) 鈭� (- 1, 2)$

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Most popular questions from this chapter

In the following exercise, find

(a) $R (x) = f (x) + g (x)$

(b) $R (x) = f (x) - g (x)$ :

role="math" localid="1644749808906" $f (x) = \frac{5}{x + 7} g (x) = \frac{10 x}{x^{2} - 49}$

Divide $\frac{11 + w}{w - 9} 梅 \frac{121 - w^{2}}{9 - w}$

Simplify $\frac{x^{2} + 7 x + 12}{x^{2} + 8 x + 16}$

In the following exercises, find the domain of each function.

$R (x) = \frac{x^{3} + 3 x^{2} - 4 x - 12}{x^{2} - 4}$

Divide the rational expression $\frac{3 y^{2} - 12 y - 63}{4 y + 3} 梅 (6 y^{2} - 42 y)$

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