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Absolute value inequalities deal with expressions where you need to find the range of values for a variable that satisfies an inequality involving absolute values. The absolute value measures how far a number is from zero on the number line, regardless of direction.
In any absolute value inequality, such as \( |x - a| \geq b \), we interpret this as asking which numbers have a distance of at least \( b \) from \( a \). This results in two potential inequalities:

• \( x - a \geq b \)
• \( -(x - a) \geq b \), which simplifies to \( x - a \leq -b \)

These separate inequalities help in understanding different regions on the number line that fulfill the original inequality. The next step is to solve each inequality separately to discover these regions.

When solving inequalities, especially absolute value inequalities, you often end up with multiple intervals that satisfy the inequality. These intervals are combined using a union to express the entire solution set.
The union of intervals allows us to denote that a solution can exist in one interval or in another. In the context of absolute value inequalities, if one part of the inequality allows solutions in one range and another part in a different range, the solution becomes a union of these two intervals.For example, after solving \( |x - 5| \geq 3 \), we arrive at two separate satisfying intervals: \([8, \infty)\) and \((-\infty, 2]\). The overall solution is the union of these:

• \( (-\infty, 2] \cup [8, \infty) \)

The union symbol \( \cup \) effectively joins these intervals, showing that any value in either interval is a valid solution to the inequality.

Solving inequalities means finding all values of a variable that make the inequality true. This involves rearranging the equation, just like solving equations, but with extra care when multiplying or dividing by negative numbers, which reverses the inequality sign.
When dealing with \(\geq\) and \(>\), you're looking for numbers greater than or equal to a certain value, while \(\leq\) and \(<\) involve finding numbers less than or equal to a value.

• For \( x - 5 \geq 3 \), solve like an equality: \( x \geq 8 \) indicates numbers 8 and above.
• For the inverted part \( 5 - x \geq 3 \), simplify and rearrange to \( x \leq 2 \), capturing the values of 2 and below.

By combining these solutions using logical operators, you find the complete solution set to the absolute value inequality. The process involves isolating \( x \) in each inequality and carefully interpreting the signs to maintain true conditions throughout.