Absolute value Knowledge-Based Study

Study notes

1. What is Absolute Value?

   The absolute value of a number describes its distance from zero along a number line. Because distance measures how far apart two locations are regardless of direction, distance is never negative.

   We denote absolute value by enclosing the number inside two vertical bars: $|x|$.

   Visualizing on a Number Line

   • The number 4 is located 4 units to the right of 0, so $|4| = 4$.
   • The number $-4$ is located 4 units to the left of 0, so $|-4| = 4$.
   • The number 0 is at zero distance from itself, so $|0| = 0$.

   Notice that opposite numbers share the exact same absolute value because they sit at identical distances from the origin.

   Formal Definition

   Formally, the absolute value function is defined piece by piece:

   $$|x| = \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0 \end{cases}$$

   When a number is negative, taking its absolute value negates the negative value, resulting in a positive number. For instance, $|-7.5| = -(-7.5) = 7.5$.

   Working with Fractions and Decimals

   The concept applies identically across all integers, fractions, and decimals:

   • Positive Rational Numbers: $\left|\frac{3}{5}\right| = \frac{3}{5}$ and $|4.18| = 4.18$
   • Negative Rational Numbers: $\left|-\frac{7}{8}\right| = \frac{7}{8}$ and $|-0.625| = 0.625$

   Fundamental Properties

   • Non-negativity: For any real number $a$, $|a| \ge 0$.
   • Symmetry: For any real number $a$, $|-a| = |a|$.
   • Identity of Zero: $|a| = 0$ if and only if $a = 0$.

   Absolute Value as a Grouping Symbol

   In algebraic expressions, absolute value bars function as grouping symbols, similar to parentheses or brackets. You must always simplify the entire expression inside the bars before evaluating the absolute value.

   Step-by-Step Example

   Evaluate the expression: $12 - |-9 + 4|$

   1. Simplify inside the bars: $-9 + 4 = -5$. The expression becomes $12 - |-5|$.
   2. Evaluate the absolute value: $|-5| = 5$. The expression becomes $12 - 5$.
   3. Perform remaining operations: $12 - 5 = 7$.

   Negation Outside the Bars

   Pay close attention to negative signs located outside the absolute value bars. An outside negative sign is applied after evaluating the absolute value:

   • $|-8| = 8$
   • $-|-8| = -(8) = -8$

   While the value inside $|\dots|$ will yield a non-negative result, an external negative sign will make the final value negative.

   Measuring Distance Between Two Points

   Absolute value provides a reliable way to calculate the geometric distance between any two numbers, $a$ and $b$, on a number line:

   $$\text{Distance} = |a - b| = |b - a|$$

   For example, to find the distance between $-3$ and $5$:

   $$|-3 - 5| = |-8| = 8$$

   Subtracting in the opposite order yields the same result: $|5 - (-3)| = |5 + 3| = |8| = 8$.

   Solving Basic Absolute Value Equations

   When solving an equation in the form $|x| = c$, consider three possibilities:

   • If $c > 0$: There are two solutions: $x = c$ or $x = -c$. For example, $|x| = 9$ means $x = 9$ or $x = -9$.
   • If $c = 0$: There is exactly one solution: $x = 0$.
   • If $c < 0$: There is no solution, because distance can never equal a negative value. For instance, $|x| = -3$ has no real solution.