Problem 25 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{3(x+2)}{4 x}+\frac{6(x-1)}{4 x}$$

Step-by-Step Solution

Answer

\(\frac{9}{4}\)

Step 1: Identify Common Denominator

Both fractions already share a common denominator, which is \(4x\). Hence, we can proceed to add the numerators directly.

Step 2: Add Numerators

Add the numerators together: \(3(x+2) + 6(x-1)\). Simplifying this, we first apply the distributive property: \(3x + 6 + 6x - 6\).

Step 3: Combine Like Terms in the Numerator

Combine like terms in the numerator: \(3x + 6x + 6 - 6 = 9x\).

Step 4: Simplify the Fraction

Write the simplified fraction \(\frac{9x}{4x}\). Here, we can cancel the \(x\) in the numerator and denominator, leaving us with \(\frac{9}{4}\).

Step 5: Verify Final Expression

Ensure each step followed the rules of fraction addition and simplification correctly. The final expression in simplest form is \(\frac{9}{4}\).

Key Concepts

Adding Fractions

Adding fractions can seem tricky, but it's simple when you know the steps. First, you need a common denominator, which is the bottom part of the fraction. Without a common denominator, you can't add fractions directly. For example, in our case, both fractions have the same denominator, which is \(4x\). This means you can add them immediately without any extra work on the denominators. Remember:

• Always check if the denominators are the same.
• If they are not, find the least common denominator (LCD) first.
• Once you have a common denominator, add the numerators (the top part).

After combining the numerators, just place the result over the common denominator.

Simplifying Fractions

Simplifying fractions makes them easier to work with. Once you have added or subtracted fractions, your next step is to simplify. This involves reducing the fraction to its smallest form.Here's the process:

• Look for common factors in the numerator and denominator.
• In our solution, after adding the fractions, we arrived at \(\frac{9x}{4x}\). Here, both the numerator and the denominator include an \(x\), which can be canceled out.
• Cancelling the \(x\) from both gives us \(\frac{9}{4}\), a simpler form.

What's left is your fraction in its simplest form. Simplifying makes it easier to interpret and use in further calculations.

Distributive Property

The distributive property is a fundamental tool in algebra that helps manage and simplify expressions. It is particularly useful when dealing with addition or subtraction inside brackets. In our example, the expression \(3(x+2) + 6(x-1)\) needed simplification. This is where the distributive property shines:

• Multiply the number outside the bracket by each term inside.
• For example, \(3(x+2)\) becomes \(3x + 6\) and \(6(x-1)\) becomes \(6x - 6\).

Once distributed, you can add or subtract like terms with ease. It's a straightforward method to ensure each term is treated equally and appropriately managed, making expressions simpler to solve or compute.