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Chapter 1: Problem 32

Determine the values represented by \(\underline{a}\). $$ |a|=a $$

Short Answer

Expert verified
\(a \geq 0\)

Step by step solution

01

Understanding the Absolute Value

The absolute value of a number is always non-negative. It represents the distance of that number from zero on the number line without considering direction. Therefore, the equation \(|a|\) implies that \(a\) must be equal to this non-negative distance.
02

Analyzing the Condition \(|a| = a\)

Since \(|a|\) represents the non-negative value of \(a\), and is equal to \(a\) in the equation \(|a| = a\), it means \(a\) itself must be non-negative. Therefore, \(a\) must be greater than or equal to zero. Mathematically, this condition can be expressed as \(a \geq 0\).

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Key Concepts

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

Non-negative numbers
Understanding non-negative numbers is crucial in mathematics. These numbers are defined as numbers that are equal to or greater than zero.
In simple terms, they include all positive numbers and zero but never include negative numbers.
Such numbers are very common in everyday situations. For example:
  • When you're counting objects like apples or cars, you can't have negative apples or negative cars in reality. Thus, the numbers will always be non-negative.
  • Savings in your bank account can be zero or positive but never negative if you don't owe money.
The main takeaway is that whenever a number is non-negative, it indicates the value is not pushing below zero. This concept is essential when we talk about absolute values, as these are always non-negative, representing true distances on a number line.
Number line
The number line is a visual representation of numbers in a straight line. It's an important tool to help us understand where numbers stand relative to each other.
Each point on the line corresponds to a unique number, with zero typically in the center.
On the number line:
  • Numbers to the right of zero are positive, moving towards infinity.
  • Numbers to the left of zero are negative, moving towards negative infinity.
The number line helps us visualize non-negative numbers, which are from zero and extending rightwards.
It illustrates not only the values themselves but also their absolute value, which is the distance from zero. For example, both 3 and -3 have an absolute value of 3, highlighting their equal distance from zero on the line.
Inequality
Inequality in mathematics helps us compare numbers or expressions, stating that they are not equal.
It uses symbols such as greater than ">", less than "<", and their variations like "greater than or equal to" (\(\geq\)) and "less than or equal to" (\(\leq\)).
For instance:
  • The statement \(a \geq 0\) is an inequality indicating "a" is a non-negative number. It can be zero or any positive value.
  • Similarly, \(a < b\) tells us "a" is less than "b".
Inequalities are vital when discussing conditions like those in absolute value equations.
They describe the range of values that satisfy a particular mathematical situation, such as finding which values of "a" make \(|a| = a\). In this case, the inequality \(a \geq 0\) pinpoints all valid solutions on the number line.

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