📘intermediate algebra review

key term - Absolute Value

Definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of whether the number is positive or negative. It is a way to measure the magnitude or size of a number without considering its sign or direction.

5 Must Know Facts For Your Next Test

  1. The absolute value of a number is always a non-negative value, as it represents the distance from zero on the number line.
  2. Absolute value can be used to compare the magnitudes of different numbers, regardless of their sign.
  3. Absolute value is denoted by vertical bars surrounding the number, such as |x|.
  4. Absolute value is an important concept in simplifying radical expressions and solving radical equations.
  5. Absolute value can be used to represent the distance between two points on the number line or in a coordinate plane.

Review Questions

  • How can absolute value be used to compare the magnitudes of different integers?
    • Absolute value can be used to compare the magnitudes of different integers by ignoring the sign of the numbers. For example, the absolute value of -5 is 5, and the absolute value of 7 is also 7. Since 5 is less than 7, we can say that the absolute value of -5 is less than the absolute value of 7, even though -5 is a negative number and 7 is a positive number. This allows us to compare the sizes or magnitudes of integers without being influenced by their signs.
  • Explain how absolute value is used to simplify radical expressions.
    • When simplifying radical expressions, absolute value is used to ensure that the radicand (the number under the radical sign) is a non-negative value. For example, to simplify the expression $\sqrt{x^2}$, we can rewrite it as $|x|$, which represents the absolute value of x. This is because the square root of a negative number is not a real number, so the absolute value is used to guarantee that the radicand is non-negative. Simplifying radical expressions using absolute value ensures that the final result is a valid, real-number solution.
  • Describe how absolute value is used to solve radical equations.
    • Solving radical equations often involves isolating the radical term and then squaring both sides of the equation to eliminate the radical. However, this process can lead to extraneous solutions, which are solutions that do not satisfy the original equation. To avoid this, absolute value is used to ensure that the solutions obtained are valid. When solving a radical equation, the absolute value of the radical expression is considered, which allows for the consideration of both the positive and negative roots. This process helps identify the true solutions to the original radical equation, while rejecting any extraneous solutions that may arise from the squaring process.