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Chapter 4: Problem 14

Determine the value of each expression. a. \(|-5-3|\) c. \(|2-6|\) b. \(|6-2|\) d. \(-2|-1+3|+|-5|\)

Short Answer

Expert verified
The values are: 8, 4, 4, 1.

Step by step solution

01

Calculate \(|-5-3|\)

First, simplify inside the absolute value: \[-5 - 3 = -8\]. Next, take the absolute value: \[|-8| = 8\].
02

Calculate \(|2-6|\)

First, simplify inside the absolute value: \[2 - 6 = -4\]. Next, take the absolute value: \[|-4| = 4\].
03

Calculate \(|6-2|\)

First, simplify inside the absolute value: \[6 - 2 = 4\]. Next, take the absolute value: \[|4| = 4\].
04

Calculate \-2| -1 + 3 | + |-5|\)

First, simplify inside the absolute value: \[-1 + 3 = 2\], \[|-5| = 5\]. Next, plug back in: \[-2|2| + 5 = -2 \times 2 + 5 = -4 + 5 = 1\].

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. It allows us to generalize arithmetic operations and solve for unknown variables.
Every algebraic expression involves a combination of constants (known values) and variables (unknown quantities). Here are the key elements:
  • Variables: Represented by letters such as \( x \) or \( y \).
  • Constants: Fixed values like 2, -5, or \( \frac{3}{4} \).
  • Operations: Include addition, subtraction, multiplication, and division.
In our exercise, we use these elements to perform operations within absolute value expressions. Ensuring you fully grasp basic algebra properties will help you solve more complex problems effortlessly.
Absolute Value Equations
Absolute value refers to the distance a number is from zero on a number line, regardless of direction. It is always non-negative. The absolute value of a number \( x \) is denoted as \( |x| \). Here are some properties of absolute values:
  • \( |a| \,=\, a \) if \( a \) is positive or zero
  • \( |a| \,=\, -a \) if \( a \) is negative
Let's apply these properties to our exercise steps:
  • Step 1: \( |-5 - 3| \,=\, |-8| \,=\, 8 \)
  • Step 2: \( |2 - 6| \,=\, |-4| \,=\, 4 \)
  • Step 3: \( |6 - 2| \,=\, 4 \)
For Step 4:
  • First, simplify inside the absolute value: \( |-1 + 3| \,=\, |2| \,=\, 2 \) and \( |-5| \,=\, 5 \)
  • Next, substitute back: \( -2|2| + 5 \,=\, -2 \times 2 + 5 \,=\, -4 + 5 \,=\, 1 \)
Knowing how to work with absolute values simplifies finding correct answers in algebraic equations.
Simplifying Expressions
Simplifying expressions means making them easier to comprehend or work with by combining like terms and reducing complexity. Here are some key tips:
  • Combine like terms: Terms that have the same variable raised to the same power. For example, \( 3x + 2x = 5x \).
  • Follow the order of operations: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right) - abbreviated as PEMDAS.
For our example, simplifying inside the absolute value comes first. Once simplified, then apply the absolute value:
  • For \( |-5-3| \), first simplify to \( -8 \), then take absolute value to get \( 8 \)
  • For \( |2-6| \), simplify to \( -4 \) first, then absolute value is \( 4 \)
  • For \( |6-2| \), simplify to \( 4 \), absolute value is \( 4 \)
  • In the final part, simplify \( |-1+3 \rightarrow 2 \) then look at absolute values in parts: \( -2|2| + 5 \,=\, -2 \times 2 + 5 \,=\, 1 \)
Practicing these methods of simplification makes solving algebraic expressions and equations more manageable and accurate.

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Most popular questions from this chapter

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