In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| + 3 = 3

Verified step by step guidance

1. Start by isolating the absolute value expression on one side of the equation. Given the equation \(|2x - 1| + 3 = 3\), subtract 3 from both sides to get \(|2x - 1| = 0\).
2. Recall that the absolute value of an expression equals zero only when the expression inside the absolute value is zero. So, set the inside equal to zero: \$2x - 1 = 0$.
3. Solve the linear equation \$2x - 1 = 0\( by adding 1 to both sides: \)2x = 1$.
4. Divide both sides by 2 to isolate \(x\): \(x = \frac{1}{2}\).
5. Since the absolute value equation reduces to a single solution, verify by substituting \(x = \frac{1}{2}\) back into the original equation to ensure it satisfies the equation.

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 any expression |A| = B, if B is non-negative, then A = B or A = -B. Understanding this is crucial for solving absolute value equations.

Isolating the Absolute Value Expression

Before solving an absolute value equation, isolate the absolute value term on one side of the equation. This often involves performing inverse operations such as addition or subtraction to simplify the equation into the form |expression| = constant.

Checking for No Solution Cases

If after isolating the absolute value, the equation takes the form |expression| = negative number, it has no solution because absolute values cannot be negative. Recognizing this helps avoid unnecessary calculations.