Problem 29
Question
Exer. 25-32: Rewrite the expression without using the absolute value symbol, and simplify the result. $$ |a-b| \text { if } a
Step-by-Step Solution
Verified Answer
If \( a < b \), then \(|a-b|=b-a\).
1Step 1: Understand Absolute Value Definition
Understand that the absolute value of a number represents its distance from zero on the number line, and is always non-negative. If \( x < 0 \), then \( |x| = -x \). If \( x >= 0 \), then \( |x| = x \).
2Step 2: Analyze the Given Condition
Analyze the condition given in the exercise, which states \( a < b \). This implies that \( a - b < 0 \).
3Step 3: Rewrite the Expression
Since \( a - b < 0 \), the absolute value \( |a-b| \) can be rewritten using the definition of absolute value as \(- (a-b) \).
4Step 4: Simplify the Rewritten Expression
Simplify the expression \(- (a-b)\), which is equivalent to \(-(a-b) = b-a\). This is because negating the expression flips the operands around.
Key Concepts
Distance on the Number LineNon-negative ValuesExpression Simplification
Distance on the Number Line
In mathematics, the number line is a fundamental concept that helps us visualize numbers and their relationships to each other. It's simply a straight line with numbers placed at even intervals along it. The concept of "distance on the number line" relates to how far apart numbers are from each other. Absolute value is a primary tool used to measure this distance.
The absolute value of a number is its distance from zero on the number line. For example, both \(|3|\) and \(|-3|\) are equal to 3 because both numbers are three steps from zero. This illustrates that distance on the number line is always a positive quantity, irrespective of direction.
When dealing with expressions such as \(|a-b|\), the absolute value helps determine the distance between points \(a\) and \(b\) on the line. If \(a
The absolute value of a number is its distance from zero on the number line. For example, both \(|3|\) and \(|-3|\) are equal to 3 because both numbers are three steps from zero. This illustrates that distance on the number line is always a positive quantity, irrespective of direction.
When dealing with expressions such as \(|a-b|\), the absolute value helps determine the distance between points \(a\) and \(b\) on the line. If \(a
Non-negative Values
One of the essential characteristics of absolute values is that they are always non-negative. Non-negative numbers are numbers that are either positive or zero, never negative, which aligns perfectly with the idea of measuring distance. Distances cannot be negative because they signify how far one point is from another or a reference point.
When working with expressions involving absolute values, it's crucial to transform results to ensure they are non-negative. For instance, if you start with \(a-b\) and determine it is negative—as in cases where \(a < b\)—you convert it using properties of absolute value, such as \(|a-b| = b-a\), thus maintaining a non-negative value.
When working with expressions involving absolute values, it's crucial to transform results to ensure they are non-negative. For instance, if you start with \(a-b\) and determine it is negative—as in cases where \(a < b\)—you convert it using properties of absolute value, such as \(|a-b| = b-a\), thus maintaining a non-negative value.
- Absolute values turn negative inputs into positive outputs.
- They assure that the calculated distances are non-negative, supporting the notion of distance on the number line.
Expression Simplification
Expression simplification involves rewriting mathematical expressions into a simpler or more easily manageable form, reducing complexity while maintaining equivalence. Simplifying expressions that contain absolute values often involves a keen understanding of the properties of absolute values themselves.
In our example, we simplify \(|a-b|\) under the condition \(a
This process exemplifies expression simplification, where understanding and applying mathematical rules and properties helps condense expressions to their simplest form.
In our example, we simplify \(|a-b|\) under the condition \(a
This process exemplifies expression simplification, where understanding and applying mathematical rules and properties helps condense expressions to their simplest form.
- Identify operations like absolute value that can be rewritten.
- Apply specific conditions given in problems—like \(a < b\) here—to guide simplification.
- Ensure all expressions are simplified to their simplest, most understandable forms.
Other exercises in this chapter
Problem 29
Express as a polynomial. $$ \left(x^{2}-3 y^{2}\right)^{2} $$
View solution Problem 29
Exer. 11-46: Simplify. $$ \left(5 x^{2} y^{-3}\right)\left(4 x^{-5} y^{4}\right) $$
View solution Problem 30
Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-3+\sqrt{-25})(8-\sqrt{-36}) $$
View solution Problem 30
Express as a polynomial. $$ \left(2 x^{2}+5 y^{2}\right)^{2} $$
View solution