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

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|4x + 2| = 5\). 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 = \frac{3}{4}$.
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 = -\frac{7}{4}$.
5. Write the solution set as the two values found: \(x = \frac{3}{4}\) and \(x = -\frac{7}{4}\).

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.

Solving Linear Equations

Solving linear equations involves isolating the variable by performing inverse operations such as addition, subtraction, multiplication, or division. This process helps find the value(s) of the variable that make the equation true.

Checking Solutions in Absolute Value Equations

After solving absolute value equations, it is important to check each solution by substituting back into the original equation to ensure they satisfy the absolute value condition, especially since absolute value equations can yield two possible solutions.