Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 70
Answer the given questions. $$\text { If } x-4<0, \text { is } \frac{x^{2}-16}{x^{3}+64}<0 ?$$
Short Answer
Expert verified
Yes, the inequality \(\frac{x^2 - 16}{x^3 + 64} < 0\) is true when \(x < 4\).
Step by step solution
01
Analyze the Condition
We start with the inequality given, \(x - 4 < 0\). This simplifies to \(x < 4\). This condition tells us about the possible range of \(x\).
02
Simplify the Expression
Consider the expression \(\frac{x^2 - 16}{x^3 + 64}\). Notice that this expression can be factored. The numerator \(x^2 - 16\) is a difference of squares, \(x^2 - 16 = (x-4)(x+4)\). The denominator \(x^3 + 64\) can be factored as a sum of cubes, \(x^3 + 64 = (x+4)(x^2 - 4x + 16)\).
03
Express as Factored Form
Using our factorization, we rewrite the expression as: \[\frac{(x-4)(x+4)}{(x+4)(x^2 - 4x + 16)}\]
04
Cancel Common Factors
In the expression, the \((x+4)\) terms can cancel, leaving \[\frac{x-4}{x^2 - 4x + 16}\]as long as \(x eq -4\), since division by zero is undefined.
05
Analyze Sign of Simplified Expression
Check for the sign of \(\frac{x-4}{x^2 - 4x + 16}\). The numerator \(x-4\) is negative because \(x < 4\). The denominator \(x^2 - 4x + 16\) is always positive for all real \(x\) since it has no real roots and opens upwards (being a quadratic with a positive leading coefficient).
06
Conclude Based on Signs
Since the numerator is negative and the denominator is always positive for the given condition \(x < 4\), the expression \(\frac{x-4}{x^2 - 4x + 16}\) is negative. Thus, the inequality \(\frac{x^2 - 16}{x^3 + 64} < 0\) is true for \(x < 4\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
The term **difference of squares** refers to a specific kind of polynomial expression that takes the form \(a^2 - b^2\). An important property of this expression is that it can be factored into the product of two binomials: \((a - b)(a + b)\). This is a crucial concept because it simplifies seemingly complex polynomials, making them easier to handle.
- In our exercise, the numerator \(x^2 - 16\) is a difference of squares since \(16 = 4^2\).
- The factorization becomes \((x - 4)(x + 4)\).
- This transformation helps us cancel similar terms in rational expressions.
Sum of Cubes
The **sum of cubes** is another type of polynomial expression, expressed as \(a^3 + b^3\). It can be factored using the formula \((a + b)(a^2 - ab + b^2)\). Understanding this can reduce complex expressions into simpler forms that are easier to solve.
- In the given problem, the denominator \(x^3 + 64\) is a sum of cubes since \(64 = 4^3\).
- We then factor it using \((x + 4)(x^2 - 4x + 16)\).
- This factorization aids in simplifying the entire rational expression, facilitating the analysis of its behavior, such as determining where it is positive or negative.
Factoring
**Factoring** is one of the most powerful tools in algebra. It involves rewriting an expression as a product of its factors, which are simpler expressions. Factoring is key in simplifying expressions and solving equations. This process makes complicated problems manageable and reveals underlying properties.
- In this example, both the numerator and denominator have been factored to identify and cancel out common terms.
- The original expression \(\frac{x^2 - 16}{x^3 + 64}\) becomes \(\frac{(x-4)(x+4)}{(x+4)(x^2 - 4x + 16)}\).
- We then further reduce it to \(\frac{x-4}{x^2 - 4x + 16}\) by canceling the \((x + 4)\) term, assuming \(x eq -4\).
Numerator and Denominator
In any rational expression, the terms **numerator** and **denominator** refer to the upper and lower parts of the fraction, respectively. Understanding how these components interact is essential, particularly in solving inequalities or determining the sign of an expression.
- In our exercise, the numerator \(x - 4\) is negative for \(x < 4\). This tells us something about its influence on the entire expression.
- The denominator \(x^2 - 4x + 16\) is a quadratic expression, always positive because it opens upward and has no real roots.
- By knowing the sign of the numerator and the constant positive sign of the denominator, we can draw conclusions about the overall sign of the rational expression \(\frac{x-4}{x^2 - 4x + 16}\).
Quadratic Expression
A **quadratic expression** is a polynomial of degree two, and it generally takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratics are fundamental in algebra due to their well-defined structure, which allows for a variety of techniques for solving them.
- In this problem, the denominator \(x^2 - 4x + 16\) is our quadratic expression.
- The expression has no real roots, meaning it does not cross the x-axis, and its graph is a parabola opening upwards.
- This property ensures that the quadratic is always positive, impacting the sign of our overall expression.
