Problem 51 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$3(x-4)$$

Step-by-Step Solution

Answer

The value of the expression is -24.

Step 1: Substitute the Value of x

The given expression is \(3(x-4)\). We need to substitute \(x = -4\) into the expression. Thus, the expression becomes \(3(-4-4)\).

Step 2: Simplify the Expression Inside the Parentheses

Now, simplify the expression inside the parentheses: \(-4-4\). This results in \(-8\). So the expression now becomes \(3(-8)\).

Step 3: Multiply the Resulting Expression

Multiply the number outside the parentheses by the number inside: \(3 imes (-8) = -24\).

Key Concepts

Understanding the Substitution Method

The substitution method is a handy tool when working with algebraic expressions. It involves replacing a variable with a given numerical value. In this exercise, you're given the expression \(3(x-4)\) and need to substitute \(x = -4\) into the expression.
This means you'll replace every instance of \(x\) with \(-4\). Take each variable in the expression and substitute it with the given value:

• Start by identifying the variable in your expression, in this case, "\(x\)".
• Next, substitute \(x = -4\) into the expression, transforming it from \(3(x-4)\) to \(3(-4-4)\).

By substituting directly, you avoid confusion and set yourself up for a straightforward simplification of the expression.

Simplifying Expressions Step-by-Step

Simplifying expressions is the process of making an expression easier to handle. When substituting values, it often involves working inside parentheses first, then moving to other operations.
In the expression \(3(-4-4)\), your first task is to simplify what’s inside the parentheses:

• Subtract \(-4\) from itself. Doing so means calculating \(-4 - 4\).
• This simplification results in \(-8\).

So, your expression is now reduced to \(3(-8)\).
Always remember to deal with operations inside parentheses first according to the order of operations (PEMDAS/BODMAS). This foundation makes further simplifications more straightforward.

Performing Multiplication with Integers

Multiplying integers is straightforward, but it’s essential to remember the rules regarding positive and negative numbers. In our exercise, once you've simplified inside the parentheses, you end up with \(3(-8)\). This is where multiplication comes into play:

• The expression now requires multiplying \(3\) by \(-8\).
• Multiply like you would with any two numbers: \(3 \times 8 = 24\).
• Because one number is negative, the product needs to be negative. Hence, \(3 \times (-8) = -24\).

Remember: when multiplying a positive number by a negative number, the result is always negative. This simple rule helps keep calculations correct in algebra.