Chapter 7: Q. 367 (page 732)

In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{x^{2} - 16} < 0$

Short Answer

Expert verified

Solution for rational inequality is $x \in (- 4, 4) .$

Step by step solution

Step 1. Given Information

Given that the rational inequality is $\frac{1}{x^{2} - 16} < 0 .$

Step 2. Algebraic Expression simplification

Take $f (x) = \frac{1}{x^{2} - 16}$

On applying formula and simplifying algebraic function,

$f (x) = \frac{1}{(x^{2} - 4^{2})} (x^{2} - 4^{2}) = (x + 4) (x - 4) ∵ (a^{2} - b^{2}) = (a + b) (a - b) f (x) = \frac{1}{(x - 4) (x + 4)}$

Step 3. Critical points

To find the critical points,

$(x - 4) = 0 x = 4 (x + 4) = 0 x = - 4$

There is an undefined polynomial function using this critical points. Thus, it will not be taken.

Divide the number line and test the polynomial expression sign in intervals.

In $\frac{1}{x^{2} - 16} < 0 in interval (- 4, 4) f (x) > 0, x \in (- 4, 4)$

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In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{x^{2} - 16} < 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Algebraic Expression simplification

Step 3. Critical points

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In the following exercises, solve each rational inequality and write the solution in interval notation.1x2-16<0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Algebraic Expression simplification

Step 3. Critical points

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Pure Maths

Geometry

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{x^{2} - 16} < 0$