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Chapter 2: Problem 14

ALGEBRA Evaluate each expression if \(a=-8\) and \(b=5\) $$$|a|-b$$

Short Answer

Expert verified
The value of the expression is 3.

Step by step solution

01

Substitute Values

First, substitute the given values of the variables into the expression. You are given that \( a = -8 \) and \( b = 5 \). The expression to evaluate is \( |a| - b \). So, substitute these values: \( |-8| - 5 \).
02

Evaluate the Absolute Value

Next, calculate the absolute value of \( a \). The absolute value of any number is its distance from zero on the number line. For \( a = -8 \), the absolute value \( |-8| = 8 \). So our expression becomes \( 8 - 5 \).
03

Subtract the Values

Finally, subtract the value of \( b \) from the absolute value of \( a \). We have \( 8 - 5 = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
Absolute value is a fundamental math concept that shows us how far a number is from zero, regardless of its direction on the number line. This means that both positive and negative numbers share the same absolute value if they have the same magnitude.

For example:
  • The absolute value of \( 5 \) is \( |5| = 5 \).
  • The absolute value of \( -5 \) is \( |-5| = 5 \).
Absolute value is used in many areas of mathematics to ensure we are focusing on the size of a number and not its direction, which can simplify calculations and interpretations. When we calculated \( |-8| \), we found it equals 8, because we are only interested in how far -8 is from zero on the number line, which is 8 units.
Substitution
Substitution in algebra involves replacing variables in an expression with their given numeric values. This step is crucial to simplify and evaluate expressions.

For our exercise, we were given that \( a = -8 \) and \( b = 5 \). We substituted these values into the expression \( |a| - b \), turning it into \( |-8| - 5 \). Substitution allows us to transform abstract algebraic expressions into concrete numbers that we can easily compute.
  • First, replace each variable with its assigned number.
  • Second, simplify the expression following mathematical rules like order of operations.
Substitution helps in understanding how different values affect an equation and is used in solving equations, inequalities, and all sorts of algebraic expressions.
Integer Operations
Integer operations refer to the basic arithmetic operations performed on whole numbers, including addition, subtraction, multiplication, and division. These operations follow certain properties like associativity, commutativity, and distributivity.

In our exercise, we executed a subtraction operation. After finding the absolute value of -8, we proceeded to subtract the integer 5, resulting in:
  • \( 8 - 5 = 3 \)
Subtracting integers involves calculating the difference between them, ensuring that the operation respects the rules of positive and negative values:
  • Subtracting a smaller number from a larger number gives a positive result.
  • Subtracting a larger number from a smaller number gives a negative result.
Understanding integer operations is necessary for evaluating and simplifying expressions in algebra and beyond.

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