Solve each absolute value inequality. 4 + |3 - x/3| ≥ 9

Verified step by step guidance

1. Rewrite the inequality by isolating the absolute value expression. Subtract 4 from both sides: |3 - x/3| ≥ 5.
2. Understand the definition of absolute value inequalities. For |A| ≥ B (where B > 0), this means A ≤ -B or A ≥ B.
3. Apply the definition to the inequality |3 - x/3| ≥ 5. This splits into two cases: (1) 3 - x/3 ≤ -5 and (2) 3 - x/3 ≥ 5.
4. Solve the first case (3 - x/3 ≤ -5): Subtract 3 from both sides to get -x/3 ≤ -8. Then multiply through by -3 (remember to reverse the inequality sign when multiplying by a negative number) to get x ≥ 24.
5. Solve the second case (3 - x/3 ≥ 5): Subtract 3 from both sides to get -x/3 ≥ 2. Then multiply through by -3 (again reversing the inequality sign) to get x ≤ -6. Combine the solutions: x ≤ -6 or x ≥ 24.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value bars.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving inequalities, especially those involving absolute values, it is important to consider the different cases that arise from the definition of absolute value, leading to multiple potential solutions.

Solving Absolute Value Inequalities

To solve an absolute value inequality, one must isolate the absolute value expression and then split the inequality into two separate cases. For example, if |A| ≥ B, it leads to two scenarios: A ≥ B or A ≤ -B. This method allows for finding all possible solutions that satisfy the original inequality, which is essential for complete problem-solving.