Problem 90 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -3(2.2-4.5)-2(1.9+4.5) $$

Step-by-Step Solution

Answer

The simplified expression is -5.9.

Step 1: Simplify the Expressions Inside the Parentheses

First, solve the arithmetic within each set of parentheses. For the first set, calculate: \( 2.2 - 4.5 = -2.3 \). For the second set, calculate: \( 1.9 + 4.5 = 6.4 \).

Step 2: Distribute and Multiply

Next, distribute the numbers outside the parentheses by multiplying with the simplified results from Step 1. For the first term: \(-3 \times (-2.3) = 6.9 \).For the second term: \(-2 \times 6.4 = -12.8 \).

Step 3: Combine the Results

Finally, combine the results from Step 2: \( 6.9 - 12.8 = -5.9 \).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about transforming complex problems into simpler, more manageable forms. It's like cleaning up clutter in a room to find what you need, quickly and easily.
Simplification often involves these steps:

• Removing parentheses by solving operations inside them first.
• Combining like terms to reduce the expression to its simplest form.

The goal is to make calculations easier while keeping the expression equivalent. This helps in not only solving but also understanding problems more effectively. Simplified expressions are less likely to lead to mistakes in further calculations. It's a crucial skill for solving algebraic equations with efficiency and accuracy.

Order of Operations

In math, the order of operations is like a recipe that tells you the exact sequence of steps to follow. Remembering the acronym PEMDAS will help you remember this sequence. PEMDAS stands for:

• P: Parentheses first
• E: Exponents (powers and roots, etc.)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

By following these steps in order, we ensure that every mathematical expression is solved consistently. For example, in our problem, you calculate everything inside the parentheses first before moving on to multiplication. Without following this order, different people might simplify an expression in different ways, leading to confusion and incorrect solutions.

Distributive Property

The distributive property is a handy tool to simplify expressions, especially when dealing with parentheses and multiplication. It allows you to distribute a multiplication across an addition or subtraction within the parentheses. This means:
If you have an expression like \( a(b + c) \), you can distribute \( a \) to both \( b \) and \( c \), resulting in \( ab + ac \).
In our example, using the distributive property, we multiplied each term inside the parentheses by the number outside. For instance, \(-3(2.2 - 4.5)\) became \(-3 \times (-2.3)\), simplifying the expression. This property helps eliminate parentheses and reveals the true value of the expression you are working with. It's a foundational concept that makes solving algebraic equations much smoother, letting you handle complex expressions with ease!