Problem 62 - Algebra A Combined Function

Question

Twice a number, added to its reciprocal is nonnegative. Find the numbers.

Step-by-Step Solution

Answer

All numbers except zero satisfy the condition.

Step 1: Define the Variables

Let's denote the unknown number by \( x \). Therefore, the expression "twice a number, added to its reciprocal" can be written as \( 2x + \frac{1}{x} \).

Step 2: Set up the Inequality

We need to find when the expression from Step 1 is nonnegative, so we set up the inequality: \[ 2x + \frac{1}{x} \geq 0. \]

Step 3: Clear the Fraction

To solve the inequality, we first eliminate the fraction by multiplying the entire inequality by \( x \) (assuming \( x eq 0 \)), yielding \[ 2x^2 + 1 \geq 0. \]

Step 4: Analyze the Inequality

The expression \( 2x^2 + 1 \) is always nonnegative since \( 2x^2 \) is always nonnegative and 1 is a positive constant. This means \( 2x^2 + 1 \geq 0 \) for all real numbers \( x \) except \( x = 0 \) (since division by zero is undefined).

Step 5: State the Solution

Therefore, the inequality is satisfied for all real numbers \( x eq 0 \). Thus, all numbers except zero satisfy the condition.

Key Concepts

Inequality Solving

Inequality solving is about finding the values of variables that satisfy an inequality. In algebra, we often deal with expressions that require us to identify where one side is greater or less than another. Understanding how to manipulate and solve these inequalities is crucial.

Firstly, recognize the difference from equations. Equations find specific solutions where both sides are equal, while inequalities identify all possible solutions that satisfy the condition. For example, to solve the inequality \[ 2x + \frac{1}{x} \geq 0, \]follow these steps:

• Identify the expression to solve: We started with `twice a number, added to its reciprocal.` This translates to \( 2x + \frac{1}{x} \).
• Set up the inequality based on the problem's condition. Here, the condition is it must be nonnegative, meaning \[ 2x + \frac{1}{x} \geq 0. \]
• Handle the inequality carefully by clearing fractions when possible. Multiply by \( x \) (assuming \( x eq 0 \)), leading to \[ 2x^2 + 1 \geq 0. \]

This transformed inequality is easy to analyze due to the nature of quadratic expressions. Quadratics like \( 2x^2 + 1 \) do not change signs easily and largely stay nonnegative.

Reciprocal Function

The reciprocal function is an important mathematical concept that involves flipping a number over. Mathematically, the reciprocal of a number \( x \) is \( \frac{1}{x} \). Think of it as dividing 1 by the number in question. It's a critical component when dealing with fractions or expressions involving division.

When working with reciprocals:

• Remember that a reciprocal switches the position of the numerator and denominator. Thus, for an integer \( n \), its reciprocal is \( \frac{1}{n} \).
• The reciprocal of a reciprocal returns you to the original number. If you take the reciprocal of \( \frac{1}{x} \), you end up back with \( x \).
• Reciprocals play a significant role when simplifying algebraic expressions, especially when involved in inequalities or complex fractions.

In our exercise, the reciprocal of \( x \) in the expression \[ 2x + \frac{1}{x}, \]means we're adding the reciprocal directly to its double.

Real Numbers

Real numbers encompass a vast set of numbers that include natural numbers, whole numbers, integers, rational, and irrational numbers. They form a continuous line on the number line and are used in everyday calculations. Understanding real numbers is crucial when solving inequalities as solutions often lie within this set.

Key characteristics of real numbers include:

• They can be positive, negative, or zero.
• Real numbers can be decimals or fractions, provided they can be pinpointed on a continuous number line.
• They include both rational numbers (like \( \frac{1}{2} \) or \( 3 \)) and irrational numbers (like \( \pi \) or \( \sqrt{2} \)).

In the given exercise, the solution states that all real numbers \( x eq 0 \) satisfy the inequality. This means every value on the real number line, except zero, will work because dividing by zero makes the expression undefined. Recognizing this restriction is essential when working with real numbers in algebraic contexts.