Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |2 x-3| \geq 2 $$

Short answer

The solution set is \( ( - \infty,1/2 ] \cup [ 5/2, \infty ) \).

Step-by-step solution

Understand the Absolute Value Inequality

The inequality is \(|2x - 3| \geq 2\). Recall that for an absolute value inequality of the form \(|A| \geq B\), it translates to two separate inequalities: \(A \geq B\) and \(A \leq -B\).

Split the Inequality

Write the two separate inequalities for \(|2x - 3| \geq 2\). Thus, \(2x - 3 \geq 2\) and \(2x - 3 \leq -2\).

Solve the First Inequality

Solve \(2x - 3 \geq 2\). Add 3 to both sides: \(2x \geq 5\). Then, divide both sides by 2: \(x \geq 5/2\).

Solve the Second Inequality

Solve \(2x - 3 \leq -2\). Add 3 to both sides: \(2x \leq 1\). Then, divide both sides by 2: \(x \leq 1/2\).

Combine the Solutions

Combine the solutions from Step 3 and Step 4. You get \(x \geq 5/2\) or \(x \leq 1/2\).

Express in Interval Notation

The solution set in interval notation is \((\infty, 1/2] \cup [5/2, \infty)\).

Graph the Solution Set

On a number line, shade all the values starting from negative infinity up to and including 1/2. Also, shade all the values starting from 5/2 to positive infinity.

Key concepts

Absolute Value Inequalities

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

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

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.