Problem 76 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ -\left|-\frac{2}{3}\right| $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{2}{3}\).

Step 1: Understand Absolute Value

Absolute value is the distance a number is from zero on the number line, disregarding any negative sign. Therefore, for any number \( x \), \( |-x| = |x| \).

Step 2: Apply Absolute Value

Simplify \(-\left| -\frac{2}{3} \right|\) by applying the absolute value. The absolute value of \(-\frac{2}{3}\) is \(\frac{2}{3}\) since it’s a distance from zero.

Step 3: Apply the Negative Sign Outside

After finding the absolute value, introduce the negative sign from the outside. Thus \(-\left| -\frac{2}{3} \right|\) becomes \(-\frac{2}{3}\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are found to the left of zero on the number line.
They are used to represent concepts like debt, loss, or decrease. In mathematics, dealing with negative numbers often involves operations such as addition, subtraction, multiplication, and division.
Understanding negative numbers is crucial for solving problems across various math topics.

• When adding a negative number, you move left on the number line.
• Subtracting a negative number is the same as adding its positive counterpart.
• Multiplying two negative numbers results in a positive number.
• Dividing two negative numbers also results in a positive number.

Negative numbers are also significant in handling absolute values, where they become positive by definition.

Simplifying Expressions

Simplifying expressions raises the idea of making complex problems easier to solve by reducing them to their simplest form.
This process can involve combining like terms, reducing fractions, and resolving operations within expressions. The simplification of expressions, especially in algebra, often determines the ease with which further mathematical operations can be performed.
Here's a step-by-step approach for the expression:

• First, resolve operations inside any brackets or absolute values.
• Apply any negative signs outside the simplified value.
• Combine and reduce any terms where possible to ensure the simplest form is achieved.

Simplification makes expressions more concise and often easier to understand, aiding in clearer problem-solving.

Number Line

A number line is a visual representation of numbers laid out on a straight line. It is a vital concept used to understand integer relationships, scales, and spacing between numbers.
On a typical number line, zero is at the center, with positive numbers extending to the right and negative numbers extending to the left. This line can stretch infinitely in both directions.
Understanding a number line helps in:

• Visualizing addition and subtraction of both positive and negative numbers.
• Locating numbers' positions relative to one another.
• Grappling with concepts such as absolute value and distance measurement.

A number line is especially important in depicting why, for instance, the absolute value of a number is its distance from zero, ignoring any direction (sign).

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols.
The basic idea of algebra involves using letters to represent numbers and express general relationships that hold true for all numbers.
Algebra is foundational in solving equations and can be applied in various ways:

• Representing real-world problems in mathematical expressions and equations.
• Simplifying expressions to solve for unknown variables.
• Using rules such as distributive, associative, and commutative properties to manipulate expressions.
• Solving linear, quadratic, and higher-order equations.

Mastering algebra requires understanding its principles and practicing its methods to manipulate and solve equations efficiently, which is a crucial skill in mathematics.