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Chapter 2: Problem 15

ALGEBRA Evaluate each expression if \(a=-8\) and \(b=5\) $$$2|a|$$

Short Answer

Expert verified
The value of the expression is 16.

Step by step solution

01

Understanding Absolute Value

The absolute value of a number is the distance from 0 to that number on the number line, regardless of direction. So, \(|a| = |-8| = 8\). This is because the absolute value turns a negative number into its positive equivalent.
02

Substitute Into the Expression

Now, substitute the value of \(|a|\) into the expression. The expression is \(2|a|\). We already found \(|a| = 8\) in Step 1. Thus, substitute to get \(2 \times 8\).
03

Calculate the Multiplication

Perform the multiplication to evaluate the expression: \(2 \times 8 = 16\). This is the final answer.

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Key Concepts

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

Understanding Absolute Value
Absolute value is a fundamental concept in algebra that might seem complex at first, but it's quite simple once you get the hang of it. The absolute value of a number represents its distance from zero on the number line. This means it transforms any negative number into a positive one, while positive numbers remain unchanged.
For instance, if you have
  • a positive number like 5, its absolute value is also 5, because it is 5 units away from zero.
  • a negative number like -8, the absolute value becomes 8.
In our example from the original problem, the absolute value of \(-8\) is 8. That's because we ignore the minus sign and simply count the steps to zero. Just remember, whether you're dealing with positive or negative numbers, the absolute value is always non-negative.
Substitution Method
The substitution method is a powerful algebraic tool used to simplify expressions by replacing variables with their actual values. Here's how it works in practice:

Imagine you have an algebraic expression like \(2|a|\) and you're given that \(a = -8\). First, you'd calculate the absolute value of \(a\), which we found is 8. Then, you substitute this value back into the expression.
In this instance, replace \(|a|\) with 8, giving you the expression \(2 \times 8\).

This method helps make problems more manageable by breaking them down into smaller, more straightforward parts. Instead of juggling the entire equation, you gradually substitute and simplify until you reach a solution.
Performing the Multiplication Process
Once you've simplified your expression using the substitution method, the next step is performing the multiplication process. This process involves basic arithmetic, which is multiplying numbers together. Let鈥檚 walk through it:
In the expression \(2 \times 8\), you have two numbers to multiply: 2 and 8. Multiplication is just repeated addition; therefore:
  • Adding 8 two times gets you 16 (8 + 8).
  • Alternatively, adding 2 eight times also results in 16.
Multiplication is a critical step in solving many algebra problems, as it simplifies expressions and helps you reach your final answer. The faster you get comfortable with handling it, the smoother your algebra journey will be!

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