Problem 85
Question
Insert either \(<,>,\) or \(=\) in the shaded area to make a true statement. $$\frac{8}{13} \div \frac{8}{13} \quad\square\quad|-1|$$
Step-by-Step Solution
Verified Answer
The correct symbol to fill in the expression to make it a true statement would be \(=\), which means the entire equation is \(\frac{8}{13} \div \frac{8}{13} = |-1|\)
1Step 1: Simplify left expression
Simplify the expression on the left-hand side. When a fraction is divided by the same fraction, the result will always be 1. Therefore, \(\frac{8}{13} \div \frac{8}{13}\) equals 1.
2Step 2: Simplify right expression
Simplify the absolute value, which is on the right-hand sight. The absolute value of any number is the non-negative value, regardless of the sign. So, the absolute value of |-1| equals 1.
3Step 3: Find the correct symbol
Since both expressions are equal to 1, the equality symbol \(=\) should be used to fill in the shaded area to make the given equation true.
Key Concepts
Absolute ValueFraction DivisionEquality
Absolute Value
Understanding the concept of absolute value is crucial in algebra. The absolute value of a number refers to its distance from zero on the number line, regardless of direction. For example, the absolute value of both \( -4 \) and \( 4 \) is \( 4 \) because each is four units away from zero.
In the exercise, we are dealing with the absolute value of \( -1 \), which is \( 1 \). This is because \( -1 \) is one unit away from zero on the number line. The symbol for absolute value is two vertical bars surrounding the number, like this: \( |x| \). This notation is read as 'the absolute value of x.' Absolute values are always non-negative because distance cannot be negative.
It's important to remember that absolute values can be applied to variables as well, such as \( |a| \), where \( a \) can be any number. The result will remove any negative sign from \( a \) and provide the positive distance from zero.
In the exercise, we are dealing with the absolute value of \( -1 \), which is \( 1 \). This is because \( -1 \) is one unit away from zero on the number line. The symbol for absolute value is two vertical bars surrounding the number, like this: \( |x| \). This notation is read as 'the absolute value of x.' Absolute values are always non-negative because distance cannot be negative.
It's important to remember that absolute values can be applied to variables as well, such as \( |a| \), where \( a \) can be any number. The result will remove any negative sign from \( a \) and provide the positive distance from zero.
Fraction Division
Fraction division is a common operation in algebra that can initially seem confusing. To divide fractions, you actually multiply by the reciprocal. The reciprocal of a fraction is obtained by flipping its numerator and denominator. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
When you divide a fraction by itself, like \( \frac{8}{13} \div \frac{8}{13} \), you're essentially multiplying it by its reciprocal. However, since the numerator and denominator are the same, their ratio is 1, which simplifies to \( \frac{8}{13} \times \frac{13}{8} = 1 \).
This operation can be seen as scaling the fraction by the factor of 1, which will always result in 1. It is a key concept students must grasp because it forms the foundation for simplifying complex algebraic expressions involving fractions.
When you divide a fraction by itself, like \( \frac{8}{13} \div \frac{8}{13} \), you're essentially multiplying it by its reciprocal. However, since the numerator and denominator are the same, their ratio is 1, which simplifies to \( \frac{8}{13} \times \frac{13}{8} = 1 \).
This operation can be seen as scaling the fraction by the factor of 1, which will always result in 1. It is a key concept students must grasp because it forms the foundation for simplifying complex algebraic expressions involving fractions.
Equality
Equality in mathematics signifies that two expressions represent the same value. When we use the symbol \( = \) between two amounts, we're asserting that they are equal. It is the very foundation of an equation.
In our exercise, we're comparing the expressions \( \frac{8}{13} \div \frac{8}{13} \) and \( |-1| \). After simplification, both sides of the equation equal 1. Thus, we can conclude that these two expressions are equal, and we correctly fill in the shaded area with the equality symbol \( = \).
Equality can also apply to more complex relationships involving variables, where the task is to find the value of the variables that make the equality true. Recognizing and demonstrating the equality between two expressions is a fundamental skill in algebra that enables you to solve equations and understand the balance between different sides of an equation.
In our exercise, we're comparing the expressions \( \frac{8}{13} \div \frac{8}{13} \) and \( |-1| \). After simplification, both sides of the equation equal 1. Thus, we can conclude that these two expressions are equal, and we correctly fill in the shaded area with the equality symbol \( = \).
Equality can also apply to more complex relationships involving variables, where the task is to find the value of the variables that make the equality true. Recognizing and demonstrating the equality between two expressions is a fundamental skill in algebra that enables you to solve equations and understand the balance between different sides of an equation.
Other exercises in this chapter
Problem 85
In Exercises \(77-96,\) simplify each algebraic expression. $$6 b-7 b$$
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Explain how to add like terms. Give an example.
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Find the value of each expression. $$-|-9-(-6)|-(-12)$$
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Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{4}{3}-\frac{3}{4}$$
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