The expression |xy| = |x| ⋅ |y| states that the absolute value of the product of two real numbers, x and y, is equal to the product of their absolute values. This property emphasizes how multiplication interacts with the concept of absolute value, ensuring that regardless of the signs of x and y, the outcome will always be a non-negative result. Understanding this relationship is crucial for solving equations and inequalities involving absolute values.
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The property |xy| = |x| ⋅ |y| holds true for any real numbers x and y, including zero.
This property is often used to simplify expressions involving absolute values in equations or inequalities.
Understanding this property can help solve problems related to distance and magnitude in various mathematical contexts.
The absolute value function is piecewise-defined, which means it behaves differently based on whether the input is positive or negative.
This property lays the groundwork for further exploration of more complex concepts like inequalities involving absolute values.
Review Questions
How does the property |xy| = |x| ⋅ |y| apply when one of the numbers is negative?
When one of the numbers x or y is negative, the absolute value ensures that we still get a non-negative product. For example, if x = -3 and y = 2, then |xy| = |-3 * 2| = | -6 | = 6. Meanwhile, |x| ⋅ |y| = | -3 | ⋅ | 2 | = 3 ⋅ 2 = 6. This shows that regardless of the signs of x and y, the equality holds true due to how absolute values are defined.
Demonstrate how you would use the property |xy| = |x| ⋅ |y| to solve an equation like |2x| = 8.
To solve the equation |2x| = 8 using this property, we first recognize that |2x| can be expressed as |2| ⋅ |x|. Thus, we rewrite it as 2|x| = 8. Dividing both sides by 2 gives us |x| = 4. To find x, we consider both possibilities: x = 4 and x = -4. This shows how we can apply the property to isolate variables in equations.
Evaluate how the understanding of |xy| = |x| ⋅ |y| enhances your problem-solving skills in dealing with inequalities involving absolute values.
Understanding the property |xy| = |x| ⋅ |y| significantly enhances problem-solving skills when handling inequalities because it allows for a systematic approach to breaking down complex expressions. When faced with an inequality such as |xy| < c, recognizing that it can be rewritten using this property lets you analyze cases where either or both variables may be negative. It creates pathways to isolate variables effectively and make logical deductions about their possible values, ultimately leading to a solution set for the inequality.
The absolute value of a real number is its distance from zero on the number line, regardless of direction, denoted as |x|.
Distance: Distance in mathematics refers to the absolute difference between two points on the number line, reflecting how far apart they are.
Real Numbers: Real numbers include all the rational and irrational numbers that can be found on the number line, encompassing both positive and negative values.