Problem 35 - Algebra and Trigonometry

Question

\(35-54\) . Perform the addition or subtraction and simplify. $$ 2+\frac{x}{x+3} $$

Step-by-Step Solution

Answer

35 - 54 equals -19, and the expression \(2 + \frac{x}{x+3}\) is already simplified.

Step 1: Interpret the Expression

First, we need to interpret the expression. We have been given two separate math problems: calculating the difference 35 - 54, and simplifying the expression \(2 + \frac{x}{x+3}\). Let's handle them separately.

Step 2: Solve the Subtraction Problem

Calculate the difference between 35 and 54: \[ 35 - 54 = -19 \]. Since 54 is greater than 35, the result will be negative.

Step 3: Simplify the Expression

The expression we need to simplify is \(2 + \frac{x}{x+3}\). Since there are no like terms or further simplifications through factoring, the expression in the current form is the simplest way to write it unless specified for a specific value of \(x\).

Key Concepts

Subtraction Problem

Subtraction often involves taking away the second number from the first. In the problem \( 35 - 54 \), we are subtracting a larger number from a smaller one. This kind of operation results in a negative number. Here's why:
When you subtract \( 54 \) from \( 35 \), imagine moving \( 54 \) steps backwards on a number line starting from zero. So, when you reach \( 35 \), you've moved backwards and still need to take additional steps backward to reach \( -19 \). Remember:

• The result of a subtraction where the subtrahend (the number being subtracted) is larger than the minuend (the initial number) is negative.
• When calculating \( 35 - 54 \), we find the distance by reversing the order (|54 - 35| = 19) and assigning a negative sign because we moved beyond zero.

Fraction Addition

Fractions represent parts of a whole. When adding fractions such as the expression \( 2 + \frac{x}{x+3} \), it can be helpful to understand how they work together.
In this expression, \( 2 \) is a whole number, while \( \frac{x}{x+3} \) is a fraction. This presents a couple of key points:

• To add fractions with different denominators, usually, we would find a common denominator. However, because \( 2 \) is a whole number, it simply adds to the fraction without needing to combine under a common denominator.
• The expression \( 2 + \frac{x}{x+3} \) already represents the simplest form unless there's a specific value for \( x \) that could simplify it further.

Always look for the possibility of simplification by finding common terms or factoring, yet if neither applies, accept the expression in its present form.

Negative Numbers

Negative numbers form a crucial component of arithmetic. They represent values less than zero. Solving problems like \( 35 - 54 = -19 \) helps us practice understanding them.
Here’s why negative numbers are important:

• They allow us to describe losses or reductions that go below a zero point, such as debts or temperatures below freezing.
• Understanding their position on the number line helps us see them as not just smaller than zero, but as values with their own magnitude directly opposite to their positive counterparts.

Working with negatives takes some getting used to. Remember that subtracting a bigger number from a smaller one will always land you in the negatives, and visualize a number line for a better grasp.