Absolute Value Formula

Absolute Value Formula

The concept of absolute value is a fundamental mathematical operation that expresses the magnitude or distance of a number from zero on the number line, regardless of its sign. The absolute value of a number $x$, denoted as $|x|$, is defined as x itself if $x\ge 0$, and as $-x$ if x<0. This ensures that the result is always non-negative. For instance, the absolute value of 5 is $5$, and the absolute value of $-5$ is also $5$. In equations and inequalities, absolute values are commonly used to express constraints or to simplify expressions involving variables with unknown signs. In geometric contexts, absolute value represents the distance between points on a coordinate plane. Learn more about absolute value function, its formula and examples

What is Absolute Value Function?

The absolute value function, denoted by |x|, is a mathematical function that gives the “absolute” or non-negative value of a real number x. In simpler terms, it measures the distance of x  from zero on the number line.

The definition of the absolute value function is:

\[ |x| = \begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases} \]

Here’s how it works:
If \( x \) is non-negative (greater than or equal to zero), then \( |x| = x \).
If \( x \) is negative, then \( |x| = -x \), which essentially negates the negative sign, making it positive.

For example:
\( |5| = 5 \) because 5 is already non-negative.
\( |-3| = 3 \) because the absolute value removes the negative sign from -3.

Absolute Value Equation

An absolute value equation is an equation that involves the absolute value function \( |x| \). These equations typically have the form:

\[ |x| = a \]

where \( a \) is a constant.

To solve an absolute value equation \( |x| = a \), you consider the definition of the absolute value function:

\[ |x| = \begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases} \]

Based on this definition, the equation \( |x| = a \) has two cases to consider:

1. \( x \geq 0 \):
In this case, \( |x| = x \). Therefore, \( x = a \).

2. \( x < 0 \):
In this case, \( |x| = -x \). Therefore, \( -x = a \), which implies \( x = -a \).

So, the solutions to the absolute value equation \( |x| = a \) are \( x = a \) and \( x = -a \).

Graph of Absolute Value Function

The graph of the absolute value function \( y = |x| \) is a V-shaped graph that intersects the y-axis at the origin (0, 0) and extends upwards and downwards symmetrically along the x-axis.

Here are the key features of the graph of \( y = |x| \):

1. Shape: The graph resembles a “V” shape, centered at the origin (0, 0).

2. Symmetry: The graph is symmetric with respect to the y-axis. This means that for any point \( (x, y) \) on the graph, the point \( (-x, y) \) is also on the graph.

3. Intersection with Axes:
The graph intersects the y-axis at \( y = 0 \) (at the origin), indicating that \( |0| = 0 \).
The graph intersects the x-axis at \( x = 0 \), where \( |x| = 0 \).

4. Behavior at \( x = 0 \):
At \( x = 0 \), \( |x| = 0 \), so the graph touches the origin (0, 0).

5. Behavior as \( x \) becomes large:
As \( x \) becomes large positively or negatively, \( |x| \) also becomes large, leading the graph to extend upwards and downwards indefinitely along the y-axis.

Solved Examples On Absolute Value Formula

Example 1: Solve the equation \( |x| = 4 \).

Solution:

To solve \( |x| = 4 \), we consider the two cases based on the definition of the absolute value function:

Case 1: \( x \geq 0 \)
\[ |x| = x \]
\[ x = 4 \]

Case 2: \( x < 0 \)
\[ |x| = -x \]
\[ -x = 4 \]
\[ x = -4 \]

Therefore, the solutions to the equation \( |x| = 4 \) are \( x = 4 \) and \( x = -4 \).

Example 2: Solve the equation \( |3x – 2| = 5 \).

Solution:

To solve \( |3x – 2| = 5 \), we consider the two cases based on the definition of the absolute value function:

Case 1: \( 3x – 2 \geq 0 \)
\[ 3x – 2 = 5 \]
\[ 3x = 7 \]
\[ x = \frac{7}{3} \]

Case 2: \( 3x – 2 < 0 \)
\[ -(3x – 2) = 5 \]
\[ -3x + 2 = 5 \]
\[ -3x = 3 \]
\[ x = -1 \]

Therefore, the solutions to the equation \( |3x – 2| = 5 \) are \( x = \frac{7}{3} \) and \( x = -1 \).

Example 3: Solve the equation \( |2y + 1| = 3 \).

Solution:

To solve \( |2y + 1| = 3 \), we consider the two cases based on the definition of the absolute value function:

Case 1: \( 2y + 1 \geq 0 \)
\[ 2y + 1 = 3 \]
\[ 2y = 2 \]
\[ y = 1 \]

Case 2: \( 2y + 1 < 0 \)
\[ -(2y + 1) = 3 \]
\[ -2y – 1 = 3 \]
\[ -2y = 4 \]
\[ y = -2 \]

Therefore, the solutions to the equation \( |2y + 1| = 3 \) are \( y = 1 \) and \( y = -2 \).