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Chapter 1: Problem 91

\(0+\frac{1}{x^{2}}=\frac{1}{x^{2}}\)

Short Answer

Expert verified
The equation is valid for any \(x\) other than 0.

Step by step solution

01

Simplify the equation

The equation is already simplified as such: \(0+\frac{1}{x^{2}}=\frac{1}{x^{2}}\).
02

Analyze the equation

The equation essentially says that \(0+\frac{1}{x^{2}}\) is equal to \(\frac{1}{x^{2}}\), which, by the rule of addition, means that the equation is true for any \(x\) other than 0 (since dividing by 0 is undefined in mathematics).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Simplification
Equation simplification is a fundamental process in algebra. It involves reducing an expression to its simplest form by performing operations and combining like terms. In the provided example, we have the equation:
  • \(0 + \frac{1}{x^{2}} = \frac{1}{x^{2}}\)
This equation is already simplified. We see that adding 0 to any term leaves the term itself unchanged. Therefore, this equation simplifies directly to its equivalent expression on the right.
Simplifying equations helps in understanding, solving, and verifying mathematical expressions efficiently. It's like cleaning an equation so that you're left with the cleanest expression possible to work with.
Undefined Values in Mathematics
Undefined values in mathematics often occur when a mathematical operation cannot be performed. A classic example is division by zero. It is crucial to recognize values that make expressions undefined for safe calculations.
  • In the equation \( \frac{1}{x^{2}} \), when \( x = 0 \), the expression becomes undefined because dividing by zero is not possible.
Each time you solve or simplify an equation, it is important to check for values that could make the equation undefined. Identifying these values is essential, as they are not part of the solution set for the equation. Avoiding division by zero helps prevent errors in calculations and theorems.
Addition Property in Algebra
The addition property in algebra states that the sum of a number and zero is the number itself. This property is useful in maintaining the equivalence of equations during manipulation.
In the example provided, we see this property in action:
  • \(0 + \frac{1}{x^{2}} = \frac{1}{x^{2}}\)
The 0 added to \( \frac{1}{x^{2}} \) doesn't change the expression, illustrating that adding zero doesn't alter the value of a term.
This property is particularly helpful when rearranging equations and maintaining equality. It simplifies the process of modeling and computing mathematical problems by eliminating superfluous elements.

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Most popular questions from this chapter

In Exercises 17 and 18, use a graphing utility to graph \(f, g\), and \(y=x\) in the same viewing window to verify geometrically that \(g\) is the inverse function of \(f\). (Be sure to restrict the domain of \(f\) properly.) f(x)=\tan x, \quad g(x)=\arctan x

Write an algebraic expression that is equivalent to the expression. $$ \tan \left(\arccos \frac{x}{2}\right) $$

$$ \arctan x=\frac{\arcsin x}{\arccos x} $$

In Exercises 59-68, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) $$ \sec [\arcsin (x-1)] $$

The height of a radio transmission tower is 70 meters, and it casts a shadow of length 30 meters (see figure). Find the angle of elevation of the sun.
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