Problem 60 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$7 x+\frac{12}{x}$$

Step-by-Step Solution

Answer

When \(x=-4\), the value of the expression is \(-31\).

Step 1: Substitute the value of x

First, we substitute the value of \(x = -4\) into the expression. The expression is \(7x + \frac{12}{x}\). Replace every \(x\) with \(-4\): \[ 7(-4) + \frac{12}{-4} \]

Step 2: Solve the multiplication

Calculate \(7(-4)\). \[ 7 \times (-4) = -28 \]

Step 3: Solve the division

Calculate \(\frac{12}{-4}\). \[ \frac{12}{-4} = -3 \]

Step 4: Combine the results

Now, add the results from Step 2 and Step 3: \[ -28 + (-3) = -31 \]

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, it's all about finding the value of an algebraic expression by substituting given numbers for the variables. In our example, we needed to evaluate the expression \(7x + \frac{12}{x}\) by replacing \(x\) with \(-4\). This substitution step is crucial as it sets the stage for all subsequent calculations. Remember, the original expression consisted of two parts: the multiplication of \(7\) with \(x\), and the division of \(12\) by \(x\). Once \(x = -4\) was substituted, our job was to apply basic arithmetic operations on these two parts.

• First, replace each \(x\) in the expression with \(-4\).
• This gives you the new expression \(7(-4) + \frac{12}{-4}\).
• Break down the expression into two straightforward calculations: multiplication and division, which are addressed separately.

Substituting correctly ensures accuracy in your resultant value. Small errors during this step, such as a missed negative sign, can drastically affect the outcome.

Negative Numbers

Negative numbers can be confusing at first, but they play a key role in mathematics. In our problem, the number \(-4\) was used in both multiplication and division. Here's how we handle negative numbers properly:

• Firstly, negative multiplied by positive is negative. So, \(7 \times (-4) = -28\). Notice that by just multiplying seven by four, you would get twenty-eight, and the negative sign in front is carried from the negative four.
• Secondly, dividing a positive number by a negative gives a negative result. Thus, for \(\frac{12}{-4}\), we get \(-3\). It’s essential to remember the sign rules during division too.

Negative numbers primarily affect the sign of our result but not the fundamental arithmetic operation. Keep practicing calculations involving negative numbers, and they'll become second nature.

Order of Operations

The order of operations is a fundamental concept that ensures expressions are evaluated the same way each time. This concept is sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). However, our example just involves multiplication and division.When confronted with the expression \(7(-4) + \frac{12}{-4}\), it's vital to evaluate it in the correct order:

• First handle multiplication and division, moving from left to right. Here, we handled \(7 \times (-4)\) and then \(\frac{12}{-4}\).
• Finally, proceed with addition or subtraction with the results from the previous operations. So, combining \(-28\) and \(-3\), you get \(-31\).

By following the order of operations, you can be certain your answer is both logical and consistent every time. This consistency is foundational in mathematics and helps avoid any ambiguity in complex calculations.