Solve each equation or inequality.
$\mid x+10\mid =\mid x-11\mid$

Verified step by step guidance

1. Recognize that the equation involves absolute values: \(|x+10| = |x-11|\). The absolute value of a number represents its distance from zero on the number line, so this equation states that the distance of \(x+10\) from zero is equal to the distance of \(x-11\) from zero.
2. Set up two cases to solve the equation, because absolute value equations can be split into cases where the expressions inside the absolute values are either equal or opposites:
3. Case 1: \(x + 10 = x - 11\)
4. Case 2: \(x + 10 = -(x - 11)\)
5. Solve each case separately by simplifying and isolating \(x\) to find the possible solutions.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any real number x, |x| equals x if x is non-negative, and -x if x is negative. Understanding this helps in rewriting and solving equations involving absolute values.

Solving Absolute Value Equations

Equations involving absolute values can be solved by considering the definition of absolute value, leading to two cases: one where the expressions inside the absolute values are equal, and one where they are opposites. This approach allows breaking down the equation into simpler linear equations.

Properties of Equality and Inequality

When solving equations, properties of equality allow manipulation of both sides to isolate variables. For absolute value equations, setting the expressions inside equal or opposite requires careful application of these properties to find all possible solutions.