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Absolute value inequalities are a bit of a mouthful, but let's break them down into simpler terms. The absolute value of a number is its distance from zero on the number line, regardless of direction. Hence, when you see something like \( |2x - 3| \geq 2 \), it means that the expression inside the absolute value is either greater than or equal to 2, or less than or equal to -2.

So, we split \( |2x - 3| \geq 2 \) into two separate inequalities: \[ 2x - 3 \geq 2 \] and \[ 2x - 3 \leq -2 \]. Solving each inequality gives you the set of solutions that make the original absolute value inequality true. Remember:

• The first inequality, \[ 2x - 3 \geq 2 \], simplifies to \[ x \geq 5/2 \].
  
  
• The second inequality, \[ 2x - 3 \leq -2 \], simplifies to \[ x \leq 1/2 \].

This means that any value of x that meets one of these conditions will satisfy the original inequality.

Interval notation is a shorthand way to describe the set of solutions to an inequality. It's a compact, elegant way to express ranges of numbers without writing out long inequalities.

For our inequalities, we ended up with two simple expressions: \[ x \geq 5/2 \] or \[ x \leq 1/2 \].

In interval notation, these solutions look like this:

• The values of x that are less than or equal to \[ 1/2 \] are written as \[ (-\infty , \ 1/2] \].


• The values of x that are greater than or equal to \[ 5/2 \] are written as \[ [5/2 , \ \infty) \].

Putting it all together, the complete solution in interval notation is \[ (-\infty, 1/2] \cup [5/2, \infty) \].

Notice the brackets and parentheses: Use a square bracket \( [ \) or \( ] \) when the endpoint is included in the set and use a parenthesis \( ( \) or \( ) \) when it's not.

Graphing inequalities on a number line helps to visually understand the solution set. Here's how you do it:

1. Draw a number line with enough points to mark your critical numbers (like \( 1/2 \) and \( 5/2 \) in our problem).
2. For \[ x \leq 1/2 \], plot a closed dot on \[ 1/2 \] (because 1/2 is included in the solution set) and draw a line extending to the left, towards negative infinity.
3. For \[ x \geq 5/2 \], plot a closed dot on \[ 5/2 \] (because 5/2 is included in the solution set) and draw a line extending to the right, towards positive infinity.

This effectively shows that all values less than or equal to \[ 1/2 \] and greater than or equal to \[ 5/2 \] satisfy the original inequality \[ |2x - 3| \geq 2 \]. The graph becomes very handy when you are dealing with more complex inequalities or systems of inequalities.