Problem 57 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$\frac{16}{x}+3 x$$

Step-by-Step Solution

Answer

The value of the expression is -16.

Step 1: Substitute -4 for x

The expression given is \( \frac{16}{x} + 3x \). First, substitute \( x = -4 \) into the expression to replace \( x \) with \( -4 \). The expression becomes \( \frac{16}{-4} + 3(-4) \).

Step 2: Evaluate the Fraction

Calculate the value of the fraction \( \frac{16}{-4} \). Divide 16 by -4 to get \(-4\).

Step 3: Multiply 3 by -4

Next, evaluate \( 3(-4) \). Multiply 3 by -4 to get \(-12\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3. Add \(-4\) from \( \frac{16}{-4} \) and \(-12\) from \( 3(-4) \) to get \(-4 + (-12) = -16\).

Key Concepts

Substitution Method

The substitution method is a technique used to evaluate algebraic expressions by replacing variables with their given values. This method is especially useful when you want to determine the numerical value of an expression for a specific value of the variable. Here’s how it works:

• Identify the variable that needs to be replaced. In our example, this variable is \(x\).
• Replace the variable with the given value in the entire expression. For example, if \(x = -4\), wherever you see \(x\) in your expression, you replace it with \(-4\).
• After substituting the values, simplify the expression using arithmetic operations to find the result.

This method helps break down complicated expressions into simpler forms and allows even beginners to work through algebraic problems with ease.

Arithmetic Operations

Arithmetic operations are fundamental calculations you perform on numbers, which include addition, subtraction, multiplication, and division. In evaluating algebraic expressions, understanding how to correctly apply these operations is crucial.For the expression \(\frac{16}{x} + 3x\) after substitution, we perform a series of arithmetic operations:

• Division: You divide 16 by the substituted value of \(x\), which in this case is \(-4\), resulting in \(-4\).
• Multiplication: Multiply 3 by \(-4\); remember the rule that multiplying two numbers with different signs gives a negative result, so the result is \(-12\).
• Addition: Once each part of the expression is calculated, add the resulting values together. Here, \(-4 + (-12) = -16\).

Mastering these operations ensures that you correctly simplify expressions and reach the right solutions efficiently.

Negative Numbers

Working with negative numbers can be tricky, but it becomes easier once you learn the rules. Negative numbers appear in expressions, like in this example when substituting values and performing arithmetic operations.Here are some basic rules to remember:

• Adding Negative Numbers: When you add two negative numbers, like \(-4\) and \(-12\), you keep the negative sign and add the absolute values: \(4 + 12 = 16\), resulting in \(-16\).
• Multiplying/Dividing Negative Numbers: When you multiply or divide two numbers where one is negative and the other is positive, the result is negative. As in \(3(-4) = -12\) or dividing \(16/-4 = -4\).

Understanding these rules helps in correctly solving expressions, especially when negative numbers are involved. Practice consistently to gain confidence in navigating through calculations with negatives.