Problem 61 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$8\left(\frac{x}{2}+5\right)$$

Step-by-Step Solution

Answer

The value of the expression is 24.

Step 1: Substitute the value of x

In the expression given, substitute the value of \(x = -4\). The expression becomes:\[8 \left( \frac{-4}{2} + 5 \right)\]

Step 2: Evaluate the division

Calculate \(\frac{-4}{2}\), which simplifies to \(-2\). The expression now looks like this:\[8(-2 + 5)\]

Step 3: Simplify inside the parentheses

Add \(-2\) and \(5\) inside the parentheses, which equals \(3\). The expression is now:\[8 \times 3\]

Step 4: Perform the multiplication

Multiply \(8\) by \(3\) to get the final result \(24\).

Key Concepts

Substitution Method

The substitution method is an essential first step when solving algebraic expressions. It involves replacing a variable with its known value so you can proceed with computations. In this exercise, we have the expression with the variable \(x\). By substituting \(x = -4\), we transform the initial algebraic form into a numerical one:

- Begin by inserting \(x = -4\) into the expression: \[ 8 \left( \frac{-4}{2} + 5 \right) \]
- This step is crucial because it simplifies the problem by removing the variable, allowing direct arithmetic calculations.

Understanding the substitution method aids in tackling various algebra problems, as it converts expressions from algebraic to arithmetic, setting a clear path for solving.

Simplifying Expressions

Once substitution is completed, simplifying the expression is the next key step. Simplification involves performing operations within the expressions to reduce it to its simplest form. Breaking it down step-by-step helps in understanding how each mathematical operation contributes to reaching the final solution.

Let's examine our substituted expression:- Evaluate the division first: \(\frac{-4}{2}\) simplifies to \(-2\).
- Having evaluated the division, our expression now looks like: \[ 8(-2 + 5) \]
- Next, resolve the addition inside the parentheses: adding \(-2\) and \(5\) results in \(3\).

Now, the expression is simplified to: \[ 8 \times 3 \]
Simplifying expressions reduces complexity and prepares the expression for final calculations.

Order of Operations

Understanding the order of operations guarantees accuracy in evaluating expressions. The widely accepted convention is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In this exercise, follow these steps:
- Begin with operations inside the parentheses. - For \(8 \left( \frac{-4}{2} + 5 \right)\), handle the operations inside first: \(\frac{-4}{2} + 5\).
- Proceed with addition within the parentheses. - Resolve it to get \(3\), simplifying it to \(8 \times 3\).
- Finally, perform multiplication: calculate \(8 \times 3\) to obtain \(24\).

Being meticulous with the order of operations ensures all calculations are performed correctly, resulting in a valid and accurate conclusion.