Problem 61 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses Then simplify the result. See Example 4 . \(-4(4 x+5)-5\)

Step-by-Step Solution

Answer

The expression simplifies to \(-16x - 25\).

Step 1: Apply Distributive Property

Use the distributive property to remove the parentheses by multiplying each term inside the parentheses by -4. The expression becomes \(-4 \times 4x + (-4) \times 5\). This simplifies to \(-16x - 20\).

Step 2: Combine Like Terms

There are no like terms to combine in \(-16x - 20 - 5\) in terms of the variable x, but you can combine the constant terms. Subtract 5 from -20, giving you \(-16x - 25\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of most algebraic operations. They are combinations of numbers, variables, and arithmetic operators such as addition, subtraction, multiplication, and division. To understand them better, consider the example \[-4(4x + 5) - 5\] This expression contains:

• A variable component: \(x\)
• Constants: -4 and -5 which represent fixed numerical values
• Arithmetic operators: Indicating operations like multiplication and subtraction

Recognizing different parts of algebraic expressions is crucial. It helps in applying algebra rules effectively, such as the distributive property, which can simplify expressions by removing parentheses and rearranging terms.

Simplifying Expressions

Simplifying expressions is a critical step towards solving algebra problems efficiently. Simplifying involves altering the expression to its simplest form, making it easier to understand and work with.

In our example, the distributive property is used initially. Starting from \[-4(4x + 5)\] you distribute -4 to both terms within the parentheses: \[-4 \times 4x + (-4) \times 5\]. This results in \[-16x - 20\]. This simplification process eliminates parentheses and reduces the complexity of the expression, making further calculations or integrations possible.

Combining Like Terms

Combining like terms is an essential technique in simplifying expressions further. It involves uniting terms which have the same variable part. Even if they have different coefficients, their variable (such as \(x\)) should match. This makes equations simpler and more organized.

In the expression \[-16x - 20 - 5\], we focus on combining constants first, which are the numbers without variables. Combining \[-20\] and \[-5\] gives \[-25\]. Therefore, the expression can be simplified to \[-16x - 25\]. Ultimately, this process helps in managing expressions more efficiently by reducing them to fewer terms, highlighting their primary components.