12 Questions
What is the definition of an absolute value inequality?
An inequality that involves an absolute value expression
What is the main difference between a simple and compound absolute value inequality?
The number of absolute value expressions
Which method involves graphing the related absolute value function on a number line?
Graphical Method
What is the property of absolute values that states |x| ≥ 0 for all x?
Property 1
What is the first step in solving an absolute value inequality using the Algebraic Method?
Isolate the absolute value expression on one side of the inequality
What is the solution set to the inequality |2x + 1| ≤ 4?
x ≥ -2.5 and x ≤ 1.5
What is the standard form of a linear equation?
Ax + By = C
How do you solve an absolute value equation?
Isolate the absolute value expression and split the equation into two separate equations
What is the first step in solving a one-variable equation?
Follow the order of operations (PEMDAS) to simplify the equation
What symbol represents 'less than or equal to' in inequality notation?
<=
How do you find the x-intercept when graphing a linear equation?
Set y = 0 and solve for x
What is the purpose of adding or subtracting the same value to both sides of a linear equation?
To isolate the variable
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].
Learn about absolute value inequalities, including simple and compound inequalities, graphical and algebraic methods for solving, and key properties. Practice solving examples with step-by-step explanations.
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