Solve each equation. |7 - 3x| = 3

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|7 - 3x| = 3\). 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: \$7 - 3x = 3\( and \)7 - 3x = -3$.
3. Solve the first equation \$7 - 3x = 3\( by isolating \)x\(: subtract 7 from both sides to get \)-3x = 3 - 7\(, then divide both sides by \)-3$.
4. Solve the second equation \$7 - 3x = -3\( similarly: subtract 7 from both sides to get \)-3x = -3 - 7\(, then divide both sides by \)-3$.
5. Write the two solutions for \(x\) obtained from the two equations as the final 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 an expression |A| = B, where B ≥ 0, the equation splits into two cases: A = B or A = -B. Understanding this is essential to solving equations involving absolute values.

Solving Linear Equations

Once the absolute value equation is split into two linear equations, each can be solved using standard algebraic methods. This involves isolating the variable by performing inverse operations such as addition, subtraction, multiplication, or division to find the solution.

Checking for Extraneous Solutions

After solving the equations derived from the absolute value, it is important to verify each solution by substituting back into the original equation. This ensures that no extraneous solutions, which do not satisfy the original absolute value equation, are included.