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Chapter 1: Problem 8

Evaluate each expression. $$|-3-2|-|2-6|$$

Short Answer

Expert verified
The value of the expression | -3 -2 | = 1

Step by step solution

01

Simplify Inside The Absolute Value Bars

Evaluate the expressions inside the absolute value bars first. For the first absolute value, We have | -3 -2 | = | -3 -2 | = -5. For the second absolute value, we have | 2-6 | = | -4
02

Subtract The Results

Finally, subtract the values obtained: 5 - 4 = 1

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Key Concepts

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

Absolute Value
Absolute value refers to the distance of a number from zero on the number line. This value is always non-negative.
For any number \( x \), the absolute value is represented as \( |x| \).

For example:
  • \( |3| = 3 \)
  • \( |-3| = 3 \)
This is because both 3 and -3 are three units away from zero, regardless of direction.
In our exercise, we apply absolute value to the expressions inside the bars before doing any other operations.
Subtraction
Subtraction is a basic arithmetic operation where we take away one number from another. It's important to keep track of negative signs during subtraction.
For the expression \( -3 - 2 \):
  • First, consider both numbers: -3 and -2.
  • Subtracting 2 from -3, we move two units left on the number line, resulting in -5.
Similarly, for the expression \( 2 - 6 \):
  • Consider both numbers: 2 and 6.
  • Subtracting 6 from 2 moves four units left, resulting in -4.
Recognizing and correctly handling negative numbers is crucial to solving these expressions.
Evaluating Expressions
Evaluating expressions involves determining the value of an algebraic expression by performing the operations in the correct order.
When evaluating expressions with absolute value, we first need to simplify inside the absolute value bars.
In our exercise, we start by simplifying:
  • Expression 1: \( -3 - 2 \)
  • Expression 2: \( 2 - 6 \)
Next, we find the absolute value of the results:
For \( -3 - 2 \):
  • \( | -5 | = 5 \)
For \( 2 - 6 \):
  • \( | -4 | = 4 \)
This allows us to correctly evaluate the overall expression.
Simplification Steps
Simplifying an expression means reducing it to its simplest form. For absolute value expressions, follow these simplification steps:

1. **Solve Inside The Absolute Value Bars**: Start by evaluating the expressions inside the absolute value bars.
  • Example: \( |-3 - 2| \) becomes \( |-5| \).
  • Example: \( |2 - 6| \) becomes \( |-4| \).
2. **Apply Absolute Value**: Convert the results to their absolute values:
  • \( |-5| = 5 \)
  • \( |-4| = 4 \)
3. **Perform Remaining Operations**: Finally, use the resulting absolute values in any remaining arithmetic operations. For our exercise, we subtract:
  • \( 5 - 4 = 1 \)
By following these steps, you ensure that the expression is simplified correctly and thoroughly. This approach tests your understanding of both absolute values and arithmetic operations.

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