In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| = 5

Verified step by step guidance

1. Recall that the absolute value equation \(|A| = B\) can be rewritten as two separate equations: \(A = B\) and \(A = -B\), provided that \(B \geq 0\).
2. Identify \(A\) and \(B\) in the given equation \(|2x - 1| = 5\). Here, \(A = 2x - 1\) and \(B = 5\).
3. Set up the two equations based on the definition of absolute value: \$2x - 1 = 5$ and \$2x - 1 = -5$.
4. Solve each equation separately: For \$2x - 1 = 5\(, add 1 to both sides and then divide by 2 to isolate \)x\(. For \)2x - 1 = -5\(, add 1 to both sides and then divide by 2 to isolate \)x$.
5. Write the two solutions you found as the solution set for 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 any expression |A| = B, where B ≥ 0, the equation splits into two cases: A = B or A = -B. Understanding this definition is essential to solving absolute value equations.

Solving Linear Equations

Once the absolute value equation is split into two linear equations, solving each involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. Mastery of these steps is necessary to find the values of x that satisfy the original equation.

Checking for No Solution

If the absolute value equals a negative number, the equation has no solution because absolute values cannot be negative. Additionally, after solving, it is important to verify solutions in the original equation to ensure they are valid and not extraneous.