Problem 86 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ \left(a^{2}+1\right)^{1 / 2} \square a+1 $$

Step-by-Step Solution

Answer

The correct symbol is \( \neq \).

Step 1: Understand the Expression

Consider the expression \( \left(a^{2}+1\right)^{1 / 2} \square a+1 \). We need to determine if replacing \( \square \) with \( = \) or \( eq \) makes the statement true. To do this, we'll analyze each side of the expression.

Step 2: Analyze the Left Side of the Expression

The left side of the expression is \( \left(a^{2}+1\right)^{1 / 2} \). This simplifies to \( \sqrt{a^2 + 1} \). Since \( a^2 \) is always non-negative, \( a^2 + 1 \) is positive, making \( \sqrt{a^2 + 1} \) a real number greater than or equal to 1.

Step 3: Analyze the Right Side of the Expression

The right side of the expression is \( a + 1 \). This depends on the value of \( a \). It will be greater than or equal to 1 if \( a \geq 0 \). For \( a < 0 \), \( a + 1 \) could be any real number.

Step 4: Compare the Expresssions for Different Values of a

For \( a \geq 0 \), \( \left( a^2 + 1 \right)^{1/2} = a + 1 \) is generally false since \( \left( a^2 + 1 \right)^{1/2} \geq \sqrt{a^2} = |a| \), and since \( |a| = a \) for \( a \geq 0 \). But always \( \sqrt{a^2 + 1} > a \). Therefore, typically \( \left( a^2 + 1 \right)^{1/2} eq a + 1 \).

Step 5: Form a Conclusion

Since the two sides of the equation involve different expressions and don't balance out for any value of \( a \), the most appropriate symbol is \( eq \). Thus, the expression \( \left(a^{2}+1\right)^{1 / 2} eq a+1 \) is valid for any real number \( a \).

Key Concepts

Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra. Think of them as sentences that tell a mathematical story. Each part of the expression contributes to its overall meaning. In the expression \( \left(a^{2}+1\right)^{1 / 2} \), we see variables \(a\), operations (addition and exponent), and the application of a square root. This expression simplifies some algebraic concepts:

• **Variables**: Symbols like \(a\) represent numbers we don’t know yet. They make us flexible as we try out different values.
• **Operations**: Here, operations include squaring \(a\) and adding 1.
• **Grouping and Order of Operations**: Parentheses and powers dictate the order in which operations occur.

To truly understand expressions, break them into parts, simplifying where possible. Look at each part independently before combining them again. Expressions can represent real-world situations, equations, or even inequalities.

Inequalities

In algebra, inequalities express a relationship where one value is not necessarily equal to another. Instead, they show if one value is larger, smaller, or different. When considering the expression \( \left(a^{2}+1\right)^{1 / 2} eq a+1 \), we deal with inequalities because we are comparing two separate mathematical ideas.

• **Symbols**: The \(eq\) symbol means "is not equal to." This tells us the expressions on either side are not the same for any real number \(a\).
• **Analysis**: By analyzing both sides, we see \( \sqrt{a^2 + 1} \) resulting in values typically greater than \(a + 1\) due to the nature of adding 1 under the square root.
• **Testing Values**: Test different numbers for \(a\) to understand how inequalities work. For example, if \(a = 0\), the left side becomes \( \sqrt{1} = 1\) while the right side becomes \(0 + 1 = 1\), which still shows that generally, \( \sqrt{a^2 + 1} > a\).

These symbols and logic help in understanding not just when things are equal, but when they differ, guiding us through a solution involving inequalities.

Square Root

The square root is a function that, for any non-negative number \(x\), produces a number \(y\) such that \(y^2 = x\). In our expression, \( \left(a^{2}+1\right)^{1 / 2} \), we interpret this as \( \sqrt{a^2 + 1} \). Understanding square roots is critical because:

• **Nature and Properties**: The result of a square root is always non-negative when starting with a positive number or zero.
• **Different from Squaring**: Squaring and square rooting undo each other. However, the square root of a squared term \(a^2\) will only yield \(|a|\), emphasizing absolute value and positive results.
• **Application**: Shows how roots can affect expressions; in \( \sqrt{a^2 + 1} \), adding 1 inside the function ensures it’s always bigger than \(|a|\), influencing how the expression \(eq\) \(a + 1\).

Square root operations define relationships in algebra, indicating how one side of an equation or expression behaves compared to another.