# 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 , denoted as , is defined as itself if , and as if . This ensures that the result is always non-negative. For instance, the absolute value of is , and the absolute value of is also . 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 $$y = -2$$.

### 1. What is an absolute value equation?

An absolute value equation is an equation that contains the absolute value function

$|x|$

. It typically takes the form ax+b=c where a, b, c are constants and x is variable

### 2. Can absolute value equations have no solution?

Yes, an absolute value equation can have no solution if the equation is inconsistent with the properties of the absolute value function or if the conditions given by the equation cannot be satisfied by any real number x.

### 3. What are the possible number of solutions to an absolute value equation?

An absolute value equation typically has two solutions, unless c = 0, in which case there is one solution. This is because the absolute value function can yield two possible values of x that satisfy the equation due to its definition involving two cases (positive and negative outcomes)

### 4. Can absolute value equations be solved graphically?

Yes, absolute value equations can be solved graphically by plotting the equations involved and finding the points where the graph intersects the horizontal line corresponding to c in ax+b=c. The x-coordinates of these points are the solutions to the equation.