Problem 73 - Elementary Algebra

Question

Is the statement \(3(5 \cdot 3-3 \cdot 5)+6 \cdot 2-3 \cdot 4<0\) true or false?

Step-by-Step Solution

Answer

Question: Is the given statement true or false: $3(5 \cdot 3 - 3 \cdot 5) + 6 \cdot 2 - 3 \cdot 4 < 0$ Answer: False

Step 1: Perform operations inside the parentheses

First, we will perform the multiplication and subtraction operations inside the parentheses: $$ 3(5 \cdot 3 - 3 \cdot 5) = 3(15 - 15) $$

Step 2: Simplify the expression

Now, we will simplify the expression: $$ 3(15 - 15) = 3 \cdot 0 $$

Step 3: Compute remaining multiplication and addition operations

Next, we will compute the remaining multiplications and additions: $$ 3 \cdot 0 + 6 \cdot 2 - 3 \cdot 4 = 0 + 12 - 12 $$

Step 4: Simplify the expression

Finally, we will simplify the expression and compare it to zero: $$ 0 + 12 - 12 = 0 $$ Since the value on the left side of the inequality is exactly zero, not less than zero, the statement is #tag_highlight# false#.

Key Concepts

Order of Operations

When working with mathematical expressions, especially those involving multiple operations like addition, subtraction, multiplication, and division, it is essential to follow the correct order of operations. This order helps ensure that everyone interprets and solves the expression in the same way. You might have heard of the acronym PEMDAS or BIDMAS, which stands for Parentheses/Brackets, Exponents/Indices, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• Always start by solving anything inside parentheses or brackets. This is the first step and takes the highest priority.
• After parentheses, look for any exponents or powers, which should be resolved next.
• Then, proceed with multiplication and division, moving from left to right. This means that within a step, if you encounter both these operations, tackle them in the order they appear.
• Finally, handle addition and subtraction, also moving from left to right.

Understanding and applying these rules will help solve complex expressions correctly. In the given exercise, following the order meant starting inside the parentheses before moving to multiplication and subtraction.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra and ensures that calculations are handled efficiently. The goal is to rewrite the expression in a more compact or clearer form without changing its value. Simplification often involves combining like terms, performing arithmetic operations, and removing any unnecessary parts of the expression.

• In the provided exercise, the simplification began by handling operations inside the parentheses: \(3(5 \cdot 3 - 3 \cdot 5)\).
• By calculating these, the result \(3(15 - 15)\) becomes zero due to subtraction.
• Once the expression inside the parentheses is simplified, further multiplication with zero yields zero, clearly demonstrating simplicity.

Remember, simplifying makes complex expressions easier to understand and evaluate. It is about removing redundancies and logically arranging terms to achieve clarity.

Evaluating Expressions

Once an expression has been simplified, it is time to evaluate it. Evaluation means computing or finding the numerical value of an expression. This final step involves performing all indicated arithmetic operations following the order of operations to reach a single numerical answer.

• In this particular exercise, after all parentheses were resolved and multiplications computed, the expression was reduced to \(0 + 12 - 12\).
• The addition and subtraction here are straightforward; adding and subtracting gives a result of zero, as \(0 + 12 = 12\) followed by \(12 - 12 = 0\).

Upon evaluating the final result, the analysis showed that the inequality was incorrect because zero is not less than zero. Carefully evaluating expressions can confirm initial predictions or reveal errors in assumptions.