Which symbol placed between the expressions creates a true sentence when $x=3$?
$12 \div x+2 x$ ____ $12 \div(x+2 x)$
We are asked to compare two algebraic expressions by substituting the value \(x = 3\) into each. The two expressions are:
Our objective is to calculate the numerical value of both expressions at \(x = 3\) and determine whether the left expression is less than, greater than, or equal to the right expression. We will adhere strictly to the order of operations (PEMDAS) throughout our calculations.
First, let's calculate the value of the left expression, \(12 \div x + 2x\), when \(x = 3\).
Substituting \(3\) for \(x\), we get:
\[12 \div 3 + 2(3)\]
Following the order of operations, we perform division and multiplication before addition:
Now, we add the results:
\[4 + 6 = 10\]
So, the value of the left expression is \(10\).
Next, let's calculate the value of the right expression, \(12 \div (x + 2x)\), when \(x = 3\).
Substituting \(3\) for \(x\), we get:
\[12 \div (3 + 2(3))\]
According to the order of operations, we must evaluate the expression inside the parentheses first:
The expression simplifies to:
\[12 \div 9\]
Converting this into a fraction and simplifying:
\[\frac{12}{9} = \frac{4}{3} \approx 1.33\]
So, the value of the right expression is \(\frac{4}{3}\).
Finally, we compare the two results:
Since \(10 > \frac{4}{3}\), the correct symbol to place between the expressions is \(>\). This corresponds to option (B). The final result is \(\boxed{>}\).
This type of comparison is essential in computer programming and spreadsheet software, where the order of operations determines the outcome of a calculation. For example, when creating a budget, writing \(=1200/4+200\) (which equals 500) gives a very different result than \(=1200/(4+200)\) (which equals approximately 5.94). Understanding how parentheses change the grouping of operations is fundamental to ensuring your calculations are accurate and your data is processed correctly.