Problem 59 - Elementary Algebra

Question

Determine whether the statements for the following problems are true or false. $$6[4+8+3(26-15)] \neq 3[7(10-4)]$$

Step-by-Step Solution

Answer

Answer: True

Step 1: Simplify the left expression

First, simplify the expression inside the parentheses: \(3(26 - 15)\). Subtract \(26 - 15\) to get \(3(11)\). Now simplify the brackets: \(6[4 + 8 + 3(11)]\). Multiply \(3\) by \(11\) to get \(33\). So we have \(6[4 + 8 + 33]\). Then, add the numbers inside the brackets to get \(6[45]\). Finally, multiply \(6\) by \(45\) to get \(270\).

Step 2: Simplify the right expression

First, simplify the expression inside the parentheses: \(7(10-4)\). Subtract \(10 - 4\) to get \(7(6)\). Then, multiply \(7\) by \(6\) to get \(42\). Multiply \(3\) by \(42\) to get \(126\).

Step 3: Compare expressions

Compare the simplified expressions. The left expression is \(270\) and the right expression is \(126\). Since \(270 \neq 126\), the statement is true. So, the given statement is true.

Key Concepts

Expressions

In algebra, expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. Consider them as phrases in math that don’t have an equal sign like equations do. When we work with expressions, our goal is usually to simplify or solve them.
For example, consider the expression in the given exercise:

• Left expression: \(6[4+8+3(26-15)]\)
• Right expression: \(3[7(10-4)]\)

Here, both sides contain numbers and operations within brackets, including parentheses. Knowing the order of operations (PEMDAS/BODMAS) is essential when evaluating these expressions.
This means handling Parentheses or Brackets first, then Exponents, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Using this order helps us systematically simplify expressions without errors.

Simplification

Simplification is a core skill in algebra and means rewriting an expression in a simpler form or reducing it to its simplest terms. In the exercise, both sides of the inequality are complex expressions that needed to be simplified.
Let’s see how it’s done:

• For the left expression \(6[4+8+3(26-15)]\): - Simplify inside the parentheses first; \((26 - 15) = 11\). - Multiply \((3 \times 11 = 33)\). - Add inside the bracket; \(4 + 8 + 33 = 45\). - Lastly, \(6 \times 45 = 270\).
• For the right expression \(3[7(10-4)]\): - Simplify inside the parentheses; \((10-4) = 6\). - Multiply \((7 \times 6 = 42)\). - Finally, \(3 \times 42 = 126\).

By following steps like these, we achieve simplified versions of expressions, making further analysis easier, such as comparing two results efficiently.

Inequalities

In algebra, inequalities are used to express the relationship between expressions that are not of equal value. They use symbols such as \(<\), \(>\), \(\leq\), \(\geq\), and \(eq\). The inequality we analyze here involves the comparison of two expressions: \(6[4+8+3(26-15)] eq 3[7(10-4)]\).
After simplifying both expressions, we found:

• The left expression equals \(270\).
• The right expression equals \(126\).

Since \(270\) is clearly not equal to \(126\), the inequality \(eq\) (not equal to) holds true, confirming the statement as true. Understanding inequalities involves knowing how to simplify expressions accurately so that one can compare them effectively.