Solve each equation or inequality.
$\mid 5x-1\mid =\mid 2x+3\mid$

Verified step by step guidance

1. Recognize that the equation involves absolute values: \(|5x - 1| = |2x + 3|\). To solve this, consider the definition of absolute value, which means the expressions inside the absolute values can be equal or opposites.
2. Set up two separate equations based on the property \(|A| = |B|\) implies \(A = B\) or \(A = -B\). So, write the two cases: \$5x - 1 = 2x + 3\( and \)5x - 1 = -(2x + 3)$.
3. Solve the first equation \$5x - 1 = 2x + 3\( by isolating \)x\(: subtract \)2x\( from both sides and add \)1\( to both sides to get all \)x$ terms on one side and constants on the other.
4. Solve the second equation \$5x - 1 = -2x - 3\( by distributing the negative sign on the right side, then isolate \)x\( by adding \)2x\( to both sides and adding \)1$ to both sides.
5. Check both solutions by substituting them back into the original equation \(|5x - 1| = |2x + 3|\) to verify that both sides are equal, confirming valid solutions.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For an expression |A| = |B|, it means either A = B or A = -B, which is essential for solving equations involving absolute values.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side to find its value. When absolute values are involved, you create two separate linear equations to solve, reflecting the positive and negative cases of the expressions inside the absolute values.

Checking Solutions in Absolute Value Equations

After solving the equations derived from absolute value expressions, it is important to verify each solution by substituting back into the original equation. This ensures that extraneous solutions, which can arise from squaring or splitting absolute values, are identified and discarded.