Question

Solve the equation. $$ |2 x+1|=x+2 $$

   Solve the equation.  
$$
|2 x+1|=x+2
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 1, Problem 21

Instant Answer

Solved by Expert Brandon Cleary

07/16/2021

Step 1

This is because the absolute value of a number can be either positive or negative. So, we have: Case 1: $2x + 1 = x + 2$ Case 2: $-2x - 1 = x + 2$ Show more…

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Solve the equation. $$ |2 x+1|=x+2 $$

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Key Concepts

Case Analysis

Case analysis refers to splitting the problem into distinct scenarios based on the sign of the expression inside the absolute value. By analyzing each case, one can remove the absolute value sign and solve the resulting equations accordingly.

Extraneous Solutions

When solving absolute value equations, it is important to check the obtained solutions in the original equation. This is because some solutions derived from the algebraic manipulations may not satisfy the conditions set by the absolute value definition, resulting in extraneous solutions.

Absolute Value

The absolute value of a number or expression is its distance from zero on the number line, which means it is always nonnegative. This definition is essential when solving equations involving absolute values because it often requires considering the nonnegative and negative cases of the inner expression separately.

Piecewise Definition

Many absolute value expressions are defined piecewise; that is, they have different expressions based on whether the input is positive, zero, or negative. This concept is crucial in solving equations as it leads to different cases in which the equation must be solved, reflecting the behavior of the absolute value function.

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