Solve each equation. | 4x + 2 | = 5

Verified step by step guidance

1. Recognize that the equation involves an absolute value expression: \(|4x + 2| = 5\). Recall that \(|A| = B\) means \(A = B\) or \(A = -B\) when \(B \geq 0\).
2. Set up two separate equations based on the definition of absolute value: \$4x + 2 = 5\( and \)4x + 2 = -5$.
3. Solve the first equation \$4x + 2 = 5\( by isolating \)x\(: subtract 2 from both sides to get \)4x = 3\(, then divide both sides by 4 to find \)x$.
4. Solve the second equation \$4x + 2 = -5\( by isolating \)x\(: subtract 2 from both sides to get \)4x = -7\(, then divide both sides by 4 to find \)x$.
5. Write the solution set as the two values of \(x\) found from the two equations, representing all solutions to the original absolute value equation.

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 an expression |A| = B, where B ≥ 0, the solutions satisfy A = B or A = -B. Understanding this is essential to split the equation into two cases for solving.

Solving Linear Equations

A linear equation involves variables raised only to the first power. To solve such equations, isolate the variable by performing inverse operations like addition, subtraction, multiplication, or division. This process is used after splitting the absolute value equation into two linear equations.

Checking for Extraneous Solutions

When solving absolute value equations, it's important to verify solutions by substituting them back into the original equation. Some solutions may not satisfy the original equation due to the nature of absolute values, so checking ensures only valid solutions are accepted.