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The absolute value of a number represents its geometric distance from $$0$$ on the number line. The absolute value of a number $$n$$ is mathematically denoted using vertical bars as $$|n|$$. Because distance cannot be negative, the absolute value of any number is always greater than or equal to zero ($$|n| \ge 0$$). For example, both $$5$$ and $$-5$$ are positioned exactly five units away from zero, meaning their absolute values are identical: $$|5| = 5$$ and $$|-5| = 5$$. This illustrates that a number and its opposite share the exact same absolute value.

Absolute Value

The absolute value of a number is never negative because it represents a physical distance on a number line, and distance cannot be negative. Therefore, $$|n| \ge 0$$ for all numbers. The only number with an absolute value equal to zero is the number $$0$$ itself, because the distance from $$0$$ to $$0$$ on the number line is exactly zero units.

Non-negative Property of Absolute Value

To simplify an expression involving absolute value, determine the distance of the number inside the bars from zero:

- $|3| = 3$, because ${3}$ is three units from ${0}$.
- $|-44| = 44$, because ${-44}$ is forty-four units from ${0}$.
- $|0| = 0$, because ${0}$ is zero units from itself.

Since absolute value measures distance, the result is always the non-negative version of the number inside the bars. For a positive number or zero, the absolute value equals the number itself; for a negative number, the absolute value equals its opposite.

Simplifying Absolute Value Expressions

When an absolute value expression contains a variable, substitute the given value and then apply the absolute value. Pay close attention to whether a negative sign sits **inside** or **outside** the absolute value bars, because the placement determines the sign of the result.

**Inside the bars — always non-negative:**
- $|x|$ when $x = -35$: Substitute to get $|-35|$. Since ${-35}$ is ${35}$ units from zero, $|-35| = 35$.
- $|-y|$ when $y = -20$: Substitute to get $|-(-20)|$. Simplify the double negative: $|-(-20)| = |20| = 20$.

**Outside the bars — always non-positive:**
- $-|u|$ when $u = 12$: Substitute to get $-|12|$. The absolute value gives ${12}$, then the outside negative makes it ${-12}$.
- $-|p|$ when $p = -14$: Substitute to get $-|-14|$. The absolute value gives ${14}$, then the outside negative makes it ${-14}$.

Notice: The expression $-|\text{variable}|$ is the negative (opposite) of a distance, so it is always zero or negative — the outside minus sign is never "absorbed" by the absolute value bars.

Evaluating $|x|$, $|-y|$, and $-|u|$ for Given Values

In a technical workshop on data analysis, a supervisor describes a measurement's deviation using the term 'absolute value.' Based on the formal mathematical definition, what does the absolute value of a number represent in relation to its position on a number line?

Defining Magnitude in Technical Reports

Based on this scenario, name the mathematical concept being used to describe these distances from zero, and state the definition of this concept in terms of the number line.

Auditing and Financial Deviations

To evaluate the absolute value of a number, assess its distance from zero on the number line. For example, considering that the number $$-5$$ is located exactly five units away from $$0$$, its absolute value evaluates to $$5$$, which is written mathematically as $$|-5| = 5$$. Likewise, the positive number $$5$$ is also positioned five units away from $$0$$, giving it an absolute value of $$5$$, written as $$|5| = 5$$. These examples illustrate that both a number and its opposite evaluate to the exact same absolute value since they share the same distance from zero.

Example of Absolute Value Evaluation

When evaluating basic absolute value equations, apply the property of absolute values to determine the solutions. For the equation $$|x| = 8$$, setting up the equivalent equations yields $$x = -8$$ or $$x = 8$$, which can be simplified as $$x = \pm 8$$. If the equation equals a negative number, such as $$|y| = -6$$, there is no possible solution because an absolute value always produces a non-negative result. When an equation is set to zero, such as $$|z| = 0$$, the only equivalent equation is $$z = 0$$ since $$-0 = 0$$, resulting in a single solution.

Example: Solving Basic Absolute Value Equations

Consider the introductory absolute value inequality $$|x| \leq 5$$. This mathematical expression asks for all numeric values whose total distance from zero evaluates to less than or equal to $$5$$. Because the distinct numbers $$-5$$ and $$5$$ predictably sit exactly five units away from zero, any arbitrary number positioned safely between them on the standard number line possesses a true distance strictly less than five units. Therefore, the absolute value inequality $$|x| \leq 5$$ logically translates to the equivalent compound inequality $$-5 \leq x \leq 5$$.

Example: Solving $$|x| \leq 5$$

To successfully solve the specific absolute value inequality $$|x| < 7$$, directly apply the established property for inequalities containing a 'less than' symbol. This fundamental algebra property correctly translates the single absolute value inequality into the equivalent compound sequence $$-7 < x < 7$$. This mathematical solution robustly represents the complete set of all numbers whose calculated distance from zero remains strictly less than $$7$$. When translated graphically onto a number line, this interval is physically denoted by placing open parentheses exclusively at the boundaries of $$-7$$ and $$7$$ while confidently shading the connecting region permanently situated between them. In formal interval notation, this uninterrupted contiguous solution set is written purely as $$(-7, 7)$$.

Example: Solving $$|x| < 7$$

To practice solving basic absolute value inequalities featuring a 'less than' sign, apply the corresponding isolation property to fully evaluate $$|x| < 9$$ and $$|x| < 1$$. For the inequality $$|x| < 9$$, the algebraic expression translates identically into the compound sequence $$-9 < x < 9$$, which describes the entire collection of values nestled between $$-9$$ and $$9$$; using standard interval notation, this translates definitively to $$(-9, 9)$$. Similarly, the smaller inequality $$|x| < 1$$ securely maps to the equivalent statement $$-1 < x < 1$$, indicating the narrow band of decimal values sandwiched exactly between $$-1$$ and $$1$$, officially documented in interval notation as $$(-1, 1)$$.

Try It: Solving $$|x| < 9$$ and $$|x| < 1$$

To solve the specific absolute value inequality $$|x| > 4$$, apply the corresponding greater than property to translate the expression into the equivalent compound sequence $$x < -4$$ or $$x > 4$$. This indicates we are seeking all numbers whose actual distance from zero is strictly greater than $$4$$ units. Graphically on a standard number line, this solution is illustrated by placing open parentheses at the $$-4$$ and $$4$$ marks to demonstrate exclusion, and clearly shading the two separate regions to the left of $$-4$$ and to the right of $$4$$. Utilizing formal interval notation, this dual solution set is documented as the union $$(-\infty, -4) \cup (4, \infty)$$. Additionally, one can intuitively verify this solution by testing arbitrary values from each section, such as $$-6$$, $$0$$, and $$7$$.

Example: Solving $$|x| > 4$$

The absolute value function is fundamentally defined by the equation $$f(x) = |x|$$. Because the absolute value can be cleanly computed for any real input, its domain is all real numbers, expressed as $$(-\infty, \infty)$$. However, since absolute value represents a number's physical distance from zero, the output can never be negative. Therefore, the range is strictly restricted to non-negative real numbers, or $$[0, \infty)$$. Geometrically, the graph of this function creates a sharp, symmetrical V-shape that consists of two linear rays converging exactly at the origin $$(0, 0)$$.

Absolute Value Function

To evaluate the mathematical expression $$-|7|$$, process the absolute value component first. The absolute value of the positive integer $$7$$ is determined by its geometric distance from zero, which correctly evaluates to $$7$$. After resolving the absolute value, apply the preceding negative sign to this positive result, yielding $$-7$$. Therefore, $$-|7| = -7$$. This example demonstrates that an external negative sign fundamentally alters the final evaluated result of an absolute value expression, transforming the normally positive distance into a negative integer.

Evaluating $$-|7|$$

To solve the introductory absolute value equation $$|x| = 5$$, one must find all numbers that make the mathematical statement true. Applying the geometric definition of absolute value, this requires finding all numbers whose distance from zero on the number line is exactly $$5$$. Because both $$5$$ and $$-5$$ are positioned exactly five units away from zero, they are both valid solutions. This means that if $$|x| = 5$$, then $$x = -5$$ or $$x = 5$$. This pair of solutions can be conveniently simplified into a single mathematical statement by writing $$x = \pm 5$$, which is read aloud as '$$x$$ is equal to positive or negative $$5$$'.

Introductory Example: Solving $$|x| = 5$$

For any algebraic expression $$u$$ and any positive real number $$a$$, the absolute value inequality $$|u| < a$$ can be translated into the compound inequality $$-a < u < a$$. Similarly, the inequality $$|u| \leq a$$ translates to $$-a \leq u \leq a$$. This algebraic property demonstrates that the solution set naturally includes all values whose underlying distance from zero is constrained to be strictly less than (or less than or equal to) $$a$$. Geometrically, this solution corresponds precisely to the continuous segment on the number line bounded securely between $$-a$$ and $$a$$.

Absolute Value Inequalities with $$<$$ or $$\leq$$

An absolute value inequality is an inequality that contains an absolute value expression. When solving an absolute value inequality, the standard principles and procedures for solving basic inequalities still apply, but it is necessary to additionally consider how the properties of the absolute value impact the mathematical work.

Absolute Value Inequality

For any algebraic expression $$u$$ and any positive real number $$a$$, the condition $$|u| = a$$ means that the mathematical statement can be rewritten as two equivalent equations: $$u = -a$$ or $$u = a$$. This property stems from the fact that we are seeking all values whose distance from zero is $$a$$, and both $$a$$ and $$-a$$ satisfy this requirement. A single statement for the solutions can also be written as $$u = \pm a$$. Remember that an absolute value cannot be a negative number.

Absolute Value Equations

For any algebraic expression $$u$$ and any positive real number $$a$$, the absolute value inequality $$|u| > a$$ translates into the equivalent compound inequality $$u < -a$$ or $$u > a$$. Similarly, the non-strict inequality $$|u| \geq a$$ translates to $$u \leq -a$$ or $$u \geq a$$. This property dictates that the mathematical solution set consists of all values whose calculated distance from zero is strictly greater than (or greater than or equal to) $$a$$. Geometrically, this solution corresponds precisely to the two distinct, non-overlapping regions on the number line extending infinitely outward in opposite directions from the boundaries $$-a$$ and $$a$$.

Absolute Value Inequalities with $$>$$ or $$\geq$$

What does the absolute value of a number represent?

Two numbers are opposites, such as $$5$$ and $$-5$$. What is true about their absolute values?

Match each absolute value expression with its distance from $$0$$ on the number line.

Which of the following defines the absolute value of a number?

Arrange the steps in the correct order to find the absolute value of a number $$n$$ using a number line.

The absolute value of a number represents its _____ from $$0$$ on the number line.

Match each absolute value term or symbol with its meaning.

The absolute value of a number is its distance from $$0$$ on the number line.

Question:
Explain absolute value using a number line. Why are $$|12|$$ and $$|-12|$$ equal?

Sample answer:
Absolute value is a number's distance from $$0$$. Distance is never negative. Both $$12$$ and $$-12$$ are $$12$$ units from $$0$$, so $$|12|=|-12|=12$$.

Key points:
- Absolute value is distance from $$0$$.
- Distance is never negative.
- $$12$$ and $$-12$$ are both $$12$$ units from $$0$$.
- $$|12|=|-12|=12$$.

Rubric:
The response should define absolute value as distance from $$0$$ and explain that $$12$$ and $$-12$$ have equal absolute values because they are the same distance from $$0$$.

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