Problem 52 - 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 x

First, we substitute the given value of \( x = -4 \) into the expression \(-3(x - 4)\). This will allow us to find the value of the expression when \( x \) is \(-4\).

Step 2: Simplify Inside the Parentheses

After substituting, the expression becomes \(-3((-4) - 4)\). Now simplify inside the parentheses: \((-4) - 4 = -8\).

Step 3: Multiply

Substitute \(-8\) into the expression, which becomes \(-3(-8)\). The operation here is multiplication: multiply \(-3\) by \(-8\), which results in 24.

Key Concepts

Simplifying Expressions

Simplifying an expression involves breaking it down into its simplest form. It's a crucial step in algebra to make solving equations easier. Begin by substituting any known variables with their given values. For instance, in the expression \(-3(x-4)\), we replace \(x\) with \(-4\). This adjustment converts the expression into \(-3((-4)-4)\). Since the expression \((-4) - 4\) simplifies to \(-8\), the entire expression is now more straightforward to handle.

Being able to simplify expressions quickly allows you to focus on solving the problem rather than getting stuck on the finer details. Remember:

• Substitute known values immediately.
• Perform any operations inside parentheses first.

These steps will help you to streamline and solve equations effectively.

Parentheses in Mathematics

Parentheses are a powerful tool in mathematics used to indicate which operations should be completed first.

In the expression \(-3(x - 4)\), it's essential to deal with what's inside the parentheses first before proceeding with other operations like multiplication. This follows the order of operations rule, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Always handle calculations inside parentheses initially.
• Reevaluate the expression after removing parentheses by simplifying the operations inside them.

For example, substituting \(x = -4\) turns \((x - 4)\) into \((-4) - 4\), which simplifies to \(-8\). Only after this simplification should you address multiplication or other operations.

Multiplication of Integers

Multiplication of integers, particularly involving negative numbers, requires attention to signs. Multiplying two negative integers results in a positive number. This rule is demonstrated in the expression \(-3(-8)\).

When two negative numbers are multiplied, like in our expression, the negatives cancel each other out, and you multiply the absolute values of the numbers. So, in our example:

• Multiply the absolute values, \(3\) and \(8\) to get \(24\).
• The negative signs cancel, resulting in a positive product.

Understanding the behavior of signs in multiplication is crucial. Always remember:

• Negative * Negative = Positive
• Negative * Positive = Negative

Being clear on these rules simplifies calculations and helps avoid mistakes.