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Questions and Answers
What is the definition of an absolute value inequality?
What is the definition of an absolute value inequality?
- An inequality that involves a polynomial expression
- An inequality that involves a linear expression
- An inequality that involves a quadratic expression
- An inequality that involves an absolute value expression (correct)
What is the main difference between a simple and compound absolute value inequality?
What is the main difference between a simple and compound absolute value inequality?
- The number of absolute value expressions (correct)
- The method used to solve it
- The number of variables involved
- The type of inequality symbol used
Which method involves graphing the related absolute value function on a number line?
Which method involves graphing the related absolute value function on a number line?
- Numerical Method
- Analytical Method
- Algebraic Method
- Graphical Method (correct)
What is the property of absolute values that states |x| ≥ 0 for all x?
What is the property of absolute values that states |x| ≥ 0 for all x?
What is the first step in solving an absolute value inequality using the Algebraic Method?
What is the first step in solving an absolute value inequality using the Algebraic Method?
What is the solution set to the inequality |2x + 1| ≤ 4?
What is the solution set to the inequality |2x + 1| ≤ 4?
What is the standard form of a linear equation?
What is the standard form of a linear equation?
How do you solve an absolute value equation?
How do you solve an absolute value equation?
What is the first step in solving a one-variable equation?
What is the first step in solving a one-variable equation?
What symbol represents 'less than or equal to' in inequality notation?
What symbol represents 'less than or equal to' in inequality notation?
How do you find the x-intercept when graphing a linear equation?
How do you find the x-intercept when graphing a linear equation?
What is the purpose of adding or subtracting the same value to both sides of a linear equation?
What is the purpose of adding or subtracting the same value to both sides of a linear equation?
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Study Notes
Absolute Value Inequalities
Definition
- An absolute value inequality is an inequality that involves an absolute value expression, such as |x|, |2x + 3|, or |x - 4|.
Types of Absolute Value Inequalities
- Simple Absolute Value Inequality: Involves a single absolute value expression, e.g., |x| > 2 or |2x + 3| ≤ 5.
- Compound Absolute Value Inequality: Involves multiple absolute value expressions, e.g., |x| + |y| ≥ 3 or |x| - |y| ≤ 2.
Solving Absolute Value Inequalities
Method 1: Graphical Method
- Graph the related absolute value function on a number line.
- Identify the intervals where the function is positive or negative.
- Determine the solution set based on the inequality symbol.
Method 2: Algebraic Method
- Isolate the absolute value expression on one side of the inequality.
- Split the inequality into two cases: one where the expression inside the absolute value is non-negative, and one where it is negative.
- Solve each case separately and combine the solutions.
Key Properties
- Property 1: |x| ≥ 0 for all x.
- Property 2: |x| = |-x| for all x.
- Property 3: |ab| = |a| |b| for all a and b.
Examples
- Solve |x - 2| > 3:
- Case 1: x - 2 ≥ 0 => x ≥ 2 => x - 2 > 3 => x > 5.
- Case 2: x - 2 < 0 => x < 2 => -(x - 2) > 3 => x < -1.
- Solution set: x > 5 or x < -1.
- Solve |2x + 1| ≤ 4:
- -4 ≤ 2x + 1 ≤ 4.
- -5 ≤ 2x ≤ 3.
- -2.5 ≤ x ≤ 1.5.
- Solution set: [-2.5, 1.5].
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