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:

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.

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1. Basics of Absolute Value Equations

The simplest example is:

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:

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

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2. General Strategy

Given an equation:

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.

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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.

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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.