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Chapter 1: Problem 22

Simplify each expression. See Example 1. $$ -\left|-\frac{4}{3}\right| $$

Short Answer

Expert verified
The simplified expression is \\(-\frac{4}{3}\\).

Step by step solution

01

Understanding Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is always non-negative. So, for any expression inside the absolute value symbol \(|x|\), the value is taken as positive or zero.
02

Calculate the Absolute Value

Identify and calculate the absolute value in the expression. Here, inside the absolute value, we have \(-\frac{4}{3}\). The absolute value of this is \[\left| -\frac{4}{3} \right| = \frac{4}{3}\]. Thus, replace \(|-\frac{4}{3}|\) with \(+\frac{4}{3}\).
03

Apply the Negative Sign Outside

Now, apply the negative sign outside the absolute value, which was originally there in the expression \(-\left| -\frac{4}{3}\right|\). After simplifying, this results in: \[-\frac{4}{3}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
Understanding absolute value is fundamental in algebra. The absolute value of a number is a measure of its distance from zero on the number line. This distance is always considered positively. Think of it like the length of a path on the number line, without worrying which direction you travel to get there.
For example, the absolute value of \(-3\) and \(3\) is the same, \(3\), because both are three units away from zero. Similarly, whether you have \(-\frac{4}{3}\) or \(+\frac{4}{3}\), the absolute value is \(+\frac{4}{3}\).
Key points to remember about absolute value:
  • Absolute value is always non-negative.
  • The symbol for absolute value is vertical bars, like this: \(|x|\).
  • Inside these bars, find the distance from zero.
Simplifying Expressions
Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form. It often requires applying basic arithmetic operations and rules, like distributing and factoring, but sometimes it involves understanding symbols like absolute value.
For this expression, \-\left| -\frac{4}{3} \right|\, the process of simplification focuses on first evaluating the absolute value, which we know is \(+\frac{4}{3}\). Then, apply any outside operations, like the negative sign here, to achieve a simplified expression: \(-\frac{4}{3}\).
Steps for simplifying expressions with absolute value:
  • Focus on the absolute value section and replace it with its positive equivalent.
  • Apply any arithmetic operations outside of the absolute value next.
  • Simplify further if possible, combining any terms appropriately.
Number Line
A number line is a simple, visual way of understanding the location and magnitude of numbers. It helps us easily grasp concepts like absolute value by representing distance. Imagine a horizontal line with zero in the center, and positive numbers extending to the right and negative numbers to the left.
To understand absolute value in this sense, locate the number in question. Calculate the distance between this number and zero while disregarding direction. For our example \(-\frac{4}{3}\), move from \-\frac{4}{3}\ to zero: this distance is \(+\frac{4}{3}\).
  • Visualizes numbers relative to zero.
  • Facilitates understanding of positive/negative sides and distance.
  • Used to easily explain absolute value and distances.

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