Problem 62 - 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

First, substitute \(x = -4\) into the expression \[-8\left(\frac{x}{2}+5\right)\]. This gives us: \[-8\left(\frac{-4}{2} + 5\right)\].

Step 2: Simplify the fraction

Compute the fraction \(\frac{-4}{2}\). Divide \(-4\) by \(2\) to get \(-2\). So, the expression becomes: \[-8\left(-2 + 5\right)\].

Step 3: Simplify inside the parentheses

Now, simplify the expression inside the parentheses. Compute \(-2 + 5\). This is equal to \(3\). The expression now is: \[-8(3)\].

Step 4: Multiply

Finally, multiply \(-8\) by \(3\). \(-8 \times 3 = -24\). This means the value of the expression is \(-24\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. In the expression \(-8\left(\frac{x}{2}+5\right)\), we see several components:

• Variable: \(x\), which can represent different values.
• Numbers: \(8, 2,\) and \(5\), constants in the expression.
• Operators: \(-, +, /\), which dictate the operations to perform.

When you substitute a value into an algebraic expression, like we did with \(x = -4\), it allows you to simplify the expression to a numerical one. This makes evaluating expressions possible and reveals how changes in the variable affect the entire expression. Understanding this substitution process is crucial for solving algebraic equations.

Arithmetic Operations

Arithmetic operations are the basic mathematical procedures used to calculate expressions and include addition, subtraction, multiplication, and division. In our example, these operations play a vital role:

• Subtraction and Addition: Inside the parenthesis, we first handled subtraction and addition, simplifying \(-2 + 5\) to get 3.
• Division: Found in the fraction \(\frac{x}{2}\), where we divided \(-4\) by \(2\) to yield \(-2\).
• Multiplication: The last step was to multiply \(-8\) by our simplified result \(3\) to arrive at \(-24\).

Each operation must be performed accurately and methodically to ensure the correct calculation of the expression. Practicing these basic operations helps in solving more complex problems.

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the sequence in which operations should be carried out to correctly solve expressions. This is often remembered by the acronym PEMDAS:

• P: Parentheses first.
• E: Exponents (ie. powers and square roots, etc.).
• MD: Multiplication and Division (left-to-right).
• AS: Addition and Subtraction (left-to-right).

In the problem given, the order was crucial:- **First**, we substituted \(x = -4\) and simplified inside the parentheses \((-2 + 5)\), according to "P" in PEMDAS.
- **Next**, we performed multiplication, following the simplified expression \(-8(3)\).
By adhering to this order, we avoided calculation errors and determined that the expression equals \(-24\). Understanding the order of operations is essential for accurately solving any mathematical expression.