Question

Solve each absolute value equation. $|x|=4$

   Solve each absolute value equation.
$|x|=4$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Elayn Martin-Gay 4th Edition

Chapter 9, Problem 27

Instant Answer

Solved by Expert Ali Soave

03/15/2020

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It is always positive or zero, but never negative. Show more…

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Solve each absolute value equation. $|x|=4$

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Key Concepts

Absolute Value

The absolute value of a number measures its distance from zero on the number line, regardless of its direction (i.e., sign). This fundamental concept ensures that the absolute value is always nonnegative and forms the basis for many areas of mathematics, including distance measurement and error analysis.

Piecewise Definition of Absolute Value

The absolute value function is defined piecewise: for any real number, it is equal to the number itself if the number is nonnegative, and equal to the opposite of the number if it is negative. This definition is critical in handling absolute value expressions in equations, as it explains why an equation like |x| = a (where a is nonnegative) naturally leads to considering two separate cases.

Solving Absolute Value Equations

When solving equations that include an absolute value, the equation is typically split into two cases corresponding to the two possible situations dictated by the piecewise definition. For a general absolute value equation |expression| = value, assuming the value is nonnegative, the solution involves setting the expression equal to both the positive and negative of that value and then solving each case separately.

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