Solve each equation or inequality. |4x-12| ≥ -3

Verified step by step guidance

1. Recognize that the inequality involves an absolute value: \(|4x - 12| \geq -3\).
2. Recall that the absolute value of any expression is always non-negative, meaning \(|4x - 12|\) is always greater than or equal to 0.
3. Since \(-3\) is less than 0, the inequality \(|4x - 12| \geq -3\) is always true for any real number \(x\).
4. Conclude that the solution to the inequality is all real numbers, as the absolute value expression is always greater than or equal to any negative number.
5. Express the solution set as \(x \in \mathbb{R}\), meaning \(x\) can be any real number.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. This concept is crucial for understanding how to manipulate equations involving absolute values, as it leads to two possible cases based on the sign of the expression inside the absolute value.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). In the context of solving inequalities, it is important to understand how to interpret and manipulate them, especially when involving absolute values, as they can lead to multiple solution sets.

Solution Sets

A solution set is the collection of all values that satisfy a given equation or inequality. When solving inequalities, particularly those involving absolute values, it is essential to determine the conditions under which the inequality holds true. In this case, since the absolute value is always non-negative, the inequality |4x - 12| ≥ -3 is always true, leading to the conclusion that the solution set includes all real numbers.