|x|, also known as the absolute value of x, represents the distance between the real number x and zero on the number line. It is a fundamental concept in mathematics that is particularly important in the context of solving absolute value inequalities.
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The absolute value of a number is always a non-negative real number, regardless of the sign of the original number.
The absolute value function, denoted as $|x|$, is defined as the distance between the real number $x$ and the origin (0) on the number line.
Solving absolute value inequalities involves finding the set of all real numbers that satisfy the given inequality, which can be done by considering the cases where the expression inside the absolute value is positive or negative.
Absolute value inequalities can be transformed into compound inequalities, which can then be solved using standard techniques for solving linear or quadratic inequalities.
The absolute value function has several important properties, such as $|x + y| leq |x| + |y|$ and $||x| - |y|| leq |x - y|$, which can be useful in solving absolute value inequalities.
Review Questions
Explain the geometric interpretation of the absolute value function |x| on the number line.
The absolute value function |x| represents the distance between the real number x and the origin (0) on the number line. Regardless of the sign of x, the absolute value always gives a non-negative real number that corresponds to the distance from x to 0. This geometric interpretation is crucial for understanding the behavior of absolute value inequalities and how to solve them.
Describe the process of solving an absolute value inequality, such as |x - 3| < 5, and explain how it can be transformed into a compound inequality.
To solve an absolute value inequality, such as |x - 3| < 5, we need to consider the two cases where the expression inside the absolute value is positive and where it is negative. This allows us to transform the absolute value inequality into a compound inequality, in this case: (x - 3 > -5) and (x - 3 < 5), which can be simplified to -8 < x - 3 < 5, or -3 < x < 8. The key is to recognize that the absolute value inequality is satisfied when the variable x falls within the range defined by the compound inequality.
Analyze the relationship between the absolute value function |x| and the properties of the real number system, and explain how these properties can be used to solve more complex absolute value inequalities, such as |2x - 1| ≤ 3 and |x + 4| > 2.
The absolute value function |x| is closely tied to the properties of the real number system, such as the triangle inequality (|x + y| ≤ |x| + |y|) and the reverse triangle inequality (||x| - |y|| ≤ |x - y|). These properties can be leveraged to solve more complex absolute value inequalities, like |2x - 1| ≤ 3 and |x + 4| > 2. For example, the first inequality can be rewritten as -3 ≤ 2x - 1 ≤ 3, which simplifies to -2 ≤ x ≤ 2. The second inequality can be transformed into x + 4 < -2 or x + 4 > 2, which gives the solution set x < -6 or x > -2. Understanding these connections between the absolute value function and the properties of real numbers is crucial for solving a wide range of absolute value inequalities.