Absolute Value Equation Calculator

Absolute Value Equation Calculator

Solve typical absolute value equations like \(|x| = a,\) \(|A x + B| = c,\) or \(|A x + B| = |C x + D|\).

Step 1: Select Equation Type & Enter Parameters

Equation: |x| = a

Typical absolute value equation forms:
1) \(|x| = a\).
2) \(|A x + B| = c\).
3) \(|A x + B| = |C x + D|\).
Additional forms may involve quadratics or multiple absolute values leading to multiple sub-cases.

Absolute Value Equation Calculator (In-Depth Explanation)

Absolute Value Equation Calculator (In-Depth Explanation)

An absolute value equation involves the mathematical concept of absolute value, denoted as $|x|$. The absolute value of a real number $x$ is defined as its distance from zero on the number line, which is always nonnegative. More formally:

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

When we talk about solving an absolute value equation, we typically have an equation of the form: $$ | \,\text{(some expression)} \,| = k, $$ where $k$ is a real number (often nonnegative). To solve for the variable inside the absolute value, we break it into separate cases based on whether the expression is positive/zero or negative.

1. Basics of Absolute Value Equations

The simplest example is:

$$|x| = k.$$

If $k < 0$, there is no solution, because absolute value cannot be negative. If $k = 0$, the only solution is $x=0$. If $k > 0$, then we typically get two solutions:

$$|x| = k \quad\Longrightarrow\quad x = k \quad\text{or}\quad x = -k.$$

For more complicated expressions, like $|\,2x+5\,| = 7$, the logic is similar, but we solve each branch of the piecewise definition.

2. General Strategy

Given an equation:

$$|\,\text{Expression}\,| = \text{Constant (or other expression)}.$$

You want to find all $x$ that satisfy this. The steps:

  1. Isolate the Absolute Value: If possible, rewrite the equation so you have something like $|A(x)| = B(x)$. If $B(x)$ is negative for some $x$, those $x$ can be rejected automatically (since absolute value can’t be equal to a negative number).
  2. Define Two Cases:
    • Case 1: $\text{Expression} \ge 0$, hence $|\,\text{Expression}\,| = \text{Expression}.$
    • Case 2: $\text{Expression} < 0$, hence $|\,\text{Expression}\,| = -(\text{Expression}).$
  3. Solve Each Case Separately: You’ll get linear or polynomial equations from each branch.
  4. Check for Extraneous Solutions: Sometimes, solutions that appear from the algebra might not actually fit the original equation (e.g., if it leads to negative values on a side that must be nonnegative). A quick check by plugging back into the original equation is important.

3. Typical Absolute Value Equation Forms

Let’s see a few common patterns:

  • Simple: $|x – 3| = 5$ translates to $(x – 3)=5$ or $(x – 3)=-5$, leading to $x=8$ or $x=-2$.
  • Linear Expressions: $|2x + 1| = 4$ translates to $2x+1=4$ or $2x+1=-4$.
  • Two Absolute Values: $|x-2| = |3x+5|$ might yield multiple sets of sub-cases. For each region of $x$ that changes the sign of $(x-2)$ or $(3x+5)$, you define an equation without absolute value, then solve. This can be a bit more involved.
  • Combining with Quadratics or Polynomials: Sometimes the expressions inside the absolute value are polynomial. The principle is the same, but the resulting sub-equations can be more complex to solve.

Example: Solve $|\,2x + 5\,| = 7$

We want $2x + 5$ to be 7 or -7 in magnitude. So:

  • Case 1: $2x + 5 = 7$. Solving: $$2x = 2 \quad\Longrightarrow\quad x=1.$$
  • Case 2: $2x + 5 = -7$. Solving: $$2x = -12 \quad\Longrightarrow\quad x=-6.$$

Both $x=1$ and $x=-6$ are valid because plugging them back in yields $|\,2(1)+5\,|=|7|=7$ and $|\,2(-6)+5\,|=|-12+5|=|-7|=7$. No contradictions appear.

Key Details:
  • No Solution Cases: If your equation forces the absolute value to equal a negative number, it’s unsolvable. Example: $|x+2|=-3$ has no real solution.
  • Check Overlaps: When you have multiple expressions in absolute values, create sub-cases based on the sign of each expression. Solve each, then confirm solutions are valid in that sub-case’s domain (sign assumptions).
  • Graphical Interpretation: $|\,f(x)\,|=g(x)$ can be visualized as the distance of $f(x)$ from zero equating to $g(x)$. Where the “V-shape” of $y=|f(x)|$ intersects $y=g(x)$ indicates solutions.

Understanding the piecewise nature of absolute value and carefully applying case-by-case logic is the key to solving absolute value equations. With practice, these steps become quite routine, especially for linear expressions.