Problem 32
Question
Exer. 25-32: Rewrite the expression without using the absolute value symbol, and simplify the result. $$ \left|-x^{2}-1\right| $$
Step-by-Step Solution
Verified Answer
The expression simplifies to \( x^2 + 1 \).
1Step 1: Understand Absolute Value
The absolute value of a number or expression, denoted by vertical bars, represents its distance from zero on the number line. It is always non-negative. For the expression \( |-x^2 - 1 | \), it equals \( x^2 + 1 \) since \(-x^2 - 1\) is always negative for all real \( x \), making the absolute value positive by flipping the sign.
2Step 2: Simplify the Expression
Replace the absolute value expression with the simplified form determined in Step 1. This results in \( x^2 + 1 \) since \(-x^2 - 1 < 0\) for all real \( x \). Thus, removing the absolute value results in changing the sign of the entire expression inside.
Key Concepts
Understanding ExpressionsWorking with Real NumbersSimplification Without Absolute Value
Understanding Expressions
An expression in mathematics is a combination of numbers, variables, and operation symbols that represents a specific value. Expressions are used extensively in algebra and other areas of mathematics to formulate equations and solve problems. The given expression, \(-x^2 - 1\), involves variables and constants combined using the operations of addition and multiplication.
Expressions can be simplified or manipulated through various algebraic techniques. The absolute value, denoted by vertical bars as in \(|-x^2 - 1|\), modifies how we interpret the expression by considering its distance from zero rather than its sign.
Being comfortable with expressions means understanding how different components interact and how operations like taking the absolute value can change their meaning or structure.
Expressions can be simplified or manipulated through various algebraic techniques. The absolute value, denoted by vertical bars as in \(|-x^2 - 1|\), modifies how we interpret the expression by considering its distance from zero rather than its sign.
Being comfortable with expressions means understanding how different components interact and how operations like taking the absolute value can change their meaning or structure.
Working with Real Numbers
Real numbers encompass all the numbers on the continuous number line, including both rational numbers (like \( rac{1}{2} \) or \(-5\)) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)). They are crucial in mathematics for representing quantities and for performing calculations in various contexts.
In the exercise, we deal with real values since \( x \) is a real number. For \(-x^2 - 1\), regardless of the real number substituted for \( x \), the expression will always result in a negative value. This is because squaring any real number results in a non-negative value, and subsequently multiplying by \(-1\) ensures the term \(-x^2\) is negative. Combining it with \(-1\) results in the whole expression being negative, which is a key aspect needed when applying absolute value principles.
The beauty of working with real numbers is their applicability in numerous situations in both theoretical and practical contexts, making them essential in understanding and solving mathematical problems.
In the exercise, we deal with real values since \( x \) is a real number. For \(-x^2 - 1\), regardless of the real number substituted for \( x \), the expression will always result in a negative value. This is because squaring any real number results in a non-negative value, and subsequently multiplying by \(-1\) ensures the term \(-x^2\) is negative. Combining it with \(-1\) results in the whole expression being negative, which is a key aspect needed when applying absolute value principles.
The beauty of working with real numbers is their applicability in numerous situations in both theoretical and practical contexts, making them essential in understanding and solving mathematical problems.
Simplification Without Absolute Value
Simplification is the process of reducing a mathematical expression to its simplest form while retaining its value. Removing absolute value effectively simplifies the computation involved. When the expression \(|-x^2 - 1|\) is evaluated, it's transformed into \(x^2 + 1\). This is due to the property that the absolute value makes all expressions non-negative by flipping any negative results to positive.
To simplify, we consider the sign of the expression inside the absolute value. Here, \(-x^2 - 1\) is always negative for real \( x \), so its absolute value counterpart is positive and equal to \(x^2 + 1\). The simplification step involves recognizing this relationship and writing the expression without absolute symbols.
Key points in simplification:
To simplify, we consider the sign of the expression inside the absolute value. Here, \(-x^2 - 1\) is always negative for real \( x \), so its absolute value counterpart is positive and equal to \(x^2 + 1\). The simplification step involves recognizing this relationship and writing the expression without absolute symbols.
Key points in simplification:
- Identify the nature (positive or negative) of the expression inside the absolute value.
- Adjust signs appropriately: negative results inside absolute values turn positive.
- Write the simplified expression by eliminating the absolute value symbols.
Other exercises in this chapter
Problem 32
Express as a polynomial. $$ (x+y)^{2}(x-y)^{2} $$
View solution Problem 32
Exer. 11-46: Simplify. $$ \left(4 a^{2} b\right)^{4}\left(\frac{-a^{3}}{2 b}\right)^{2} $$
View solution Problem 33
Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{\sqrt{-36} \sqrt{-49}}{\sqrt{-16}} $$
View solution Problem 33
Express as a polynomial. $$ (\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y}) $$
View solution