Algebra 1 Practice - Solve and Graph an Absolute Value Inequality (Example 3)

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Algebra 1 Practice: Solve and Graph an Absolute Value Inequality
In Algebra 1, solving and graphing an absolute value inequality involves finding the range of values for a variable that satisfies an inequality containing absolute value expressions. Here's a step-by-step guide to tackle this:
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#### *Steps to Solve an Absolute Value Inequality*
1. *Identify the Absolute Value Inequality:*
- Recognize the form of the inequality, typically presented as \( |ax + b| \) is less than \( c \) or \( |ax + b| \) is greater than \( c \), where \( a \), \( b \), and \( c \) are constants.
2. *Isolate the Absolute Value Expression:*
- Isolate the absolute value expression by considering two cases: \( ax + b \) is positive or \( ax + b \) is negative.
3. *Solve Each Case:*
- Solve the inequality for each case separately.
4. *Combine Solutions:*
- Combine the solutions from both cases to form the complete solution set.
5. *Graph the Solution Set:*
- Graph the solution set on a number line.
6. *Example:*
- Solve and graph \( |2x - 3| \) is less than 5:
1. Isolate the absolute value expression: \( -5 \) is less than \( 2x - 3 \) is less than 5.
2. Consider two cases:
- Case 1: \( 2x - 3 \) is positive: \( -5 \) is less than \( 2x - 3 \) is less than 5.
- Case 2: \( 2x - 3 \) is negative: \( -5 \) is less than \( -(2x - 3) \) is less than 5.
3. Solve each case separately.
4. Combine the solutions to form the complete solution set.
5. Graph the solution set on a number line.
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#### *Complete Example: Solving and Graphing*
- *Absolute Value Inequality:* \( |2x - 3| \) is less than 5
- *Solution:*
- Isolate the absolute value expression: \( -5 \) is less than \( 2x - 3 \) is less than 5.
- Consider two cases:
- Case 1: \( 2x - 3 \) is positive: \( -5 \) is less than \( 2x - 3 \) is less than 5.
- Case 2: \( 2x - 3 \) is negative: \( -5 \) is less than \( -(2x - 3) \) is less than 5.
- Solve each case separately.
- Combine the solutions to form the complete solution set.
- *Graph:*
- Graph the solution set on a number line.
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By following these steps, you can effectively solve and graph absolute value inequalities, visually representing the solution set on a number line. This process helps in understanding and interpreting the solutions for various absolute value inequality problems in Algebra 1. Regular practice with different types of absolute value inequalities will help reinforce these skills.
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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa
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