Problem 78 - Intermediate Algebra

Question

Find the value of each expression. $$ -|1 . \overline{27}| $$

Step-by-Step Solution

Answer

The value is \(-1.3\overline{6}\).

Step 1: Expression

Find the value of each expression.
$$
-|1 . \overline{27}|
$$

Step 2: Apply Rules

Apply appropriate mathematical rules step by step.

Step 3: Result

The value is \(-1.3\overline{6}\).

Key Concepts

Understanding Absolute Value

Absolute value refers to the non-negative value of a number, regardless of whether the number itself is positive or negative.

This concept can be thought of as the distance a number is from zero on a number line.
For instance, the absolute value of both 3 and -3 is 3, because they are both three units away from zero.
In mathematical terms, the absolute value of a number \( x \) is represented as \( |x| \).

Here are some crucial points:

• The absolute value of 0 is 0, because it is at zero distance from zero.
• The absolute value \(|x|\) always results in a non-negative value.
• When you see the negative sign in front of the absolute value operation, such as in our exercise, you are taking the negative of the non-negative absolute value.

This is important for solving expressions like \(-|1.\overline{27}|\), where it's necessary to compute the absolute value first, and then apply the negative sign.

To find the absolute value of a repeating decimal, focus on the value without considering its sign. This ensures a clear calculation.

Working with Repeating Decimals

Repeating decimals are numbers that have a digit or a group of digits that repeat infinitely.
For example, in the case of \(1.\overline{27}\), the digits '27' repeat infinitely.
Repeating decimals often appear in problems involving fractions.
To convert a repeating decimal into a fraction, you need to follow a specific procedure:

• Identify the repeating part of the decimal.
• Set the repeating decimal equal to a variable like \( x \).
• Multiply the equation by a power of 10 that matches the cycle of the repeat to shift the decimal place up.
• Subtract the original equation from this new equation to eliminate the repeating part and solve for \( x \).

For example, if you let \( x = 1.\overline{27} \), multiplying both sides by 100 would allow you to remove the repeating part via subtraction.

Understanding how to manipulate repeating decimals is key for expressions that require exact fractional answers or further algebraic operations.

Remember, in any expression, always resolve repeated decimals by understanding their structure before proceeding with other operations like taking an absolute value.

Handling Negative Numbers

Negative numbers are numbers that are less than zero.
They are typically represented with a minus sign, \(-\).
Understanding how to work with negative numbers is vital as they frequently appear in various algebraic expressions.

Here’s how negative numbers often interact within calculations:

• Performing operations with negative numbers requires careful attention to signs. When multiplying or dividing, two negative numbers result in a positive product, but one positive and one negative result in a negative product.
• In the context of absolute values, when there is a negative sign outside of an absolute value, it reverses the sign of the resulting non-negative value. For instance, in \(-|1.\overline{27}|\), calculate the absolute value first and then apply the negative sign.
• It's crucial in solving equations and inequalities to switch the inequality sign when multiplying or dividing both sides by a negative number.

With these basics covered, you can confidently tackle expressions involving negative numbers, including those involving absolute values and repeating decimals.

Practice is essential to become proficient in managing negative values, particularly in more complex algebraic settings.