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 complete solution set to 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 must be solved separately. This involves isolating the variable by performing inverse operations such as addition, subtraction, multiplication, or division to find the value of x.

Checking Solutions in Absolute Value Equations

After finding potential solutions, it is important to verify them by substituting back into the original absolute value equation. This ensures that extraneous solutions, which can arise from the splitting process, are identified and discarded if they do not satisfy the original equation.