Problem 45 - Prealgebra

Question

Simplify each expression. $$-2(x+5)$$

Step-by-Step Solution

Answer

The simplified expression is \(-2x - 10\).

Step 1: Distribute the Negative Sign

The expression given is \(-2(x + 5)\). To simplify it, we need to distribute the negative sign across the terms inside the parenthesis. Multiply \(-2\) with each term inside the bracket:- Multiply \(-2\) with \(x\): \(-2 \times x = -2x\).- Multiply \(-2\) with \(5\):\(-2 \times 5 = -10\).

Step 2: Combine the Terms

Now, write the expression by combining the results from Step 1:The expression becomes \(-2x - 10\).There are no like terms, so this is the simplified form of the expression.

Key Concepts

Simplifying Expressions

When faced with mathematical expressions, we often need to simplify them to make them easier to work with. Simplifying expressions means taking a complex expression and reducing it to its simplest form without changing its value. This is an important concept in prealgebra and helps in understanding and solving equations.
To simplify expressions:

• Look for like terms, which are terms that have the same variable raised to the same power. Combine these terms, if applicable.
• Apply mathematical operations, like addition and subtraction, to simplify the expression.
• Remove unnecessary parentheses by applying operations inside them.

In the case of the expression \(-2(x + 5)\), we have simplified it by distributing the terms inside the parentheses, resulting in \(-2x - 10\). No further simplification is possible since there are no like terms.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions, especially when dealing with parentheses. This property states that multiplying a number by a sum of numbers inside parentheses is the same as multiplying the number by each term inside the parentheses and adding the products.
The general form of the distributive property is:\(a(b + c) = ab + ac\).
Here, \(a\) is distributed to both \(b\) and \(c\). This means that you multiply \(a\) by \(b\) and\(a\) by \(c\), then add the results.For example, in the expression \(-2(x + 5)\), we apply the distributive property by:

• Multiplying \(-2\) with \(x\) to get \(-2x\)
• Multiplying \(-2\) with \(5\) to get \(-10\)

The final expression after applying the distributive property is \(-2x - 10\).
Using this property allows for expressions to be presented in a simpler and more manageable form.

Negative Numbers

Negative numbers can sometimes be confusing, especially when they are mixed with operations like multiplication or subtraction. However, understanding how to work with negative numbers is crucial in algebra and prealgebra.
Negative numbers are less than zero and are written with a minus sign (\(-\)). When multiplying or dividing:

• A negative number multiplied by a positive number produces a negative number (e.g., \(-2 \times 5 = -10\)).
• A negative number multiplied by another negative number yields a positive number (e.g., \(-2 \times -3 = 6\)).

This is because you're multiplying two negative directions, which results in a positive direction.
In the given expression \(-2(x + 5)\), we use the rule that a negative times a positive results in a negative. This explains the terms \(-2x\) and \(-10\) in the simplified expression.