Problem 75 - Understanding Elementary Algebra with Geometry

Question

Combine: \(\frac{x}{3}-\frac{x-1}{4}+\frac{x+1}{2}\)

Step-by-Step Solution

Answer

\(\frac{7x + 9}{12}\)

Step 1 - Find the Common Denominator

Identify the denominators in the expression: 3, 4, and 2. The least common multiple (LCM) of these numbers is 12. Therefore, the common denominator is 12.

Step 2 - Rewrite Each Fraction with the Common Denominator

Convert each fraction to have the common denominator of 12. \(\frac{x}{3} = \frac{4x}{12}\), \(\frac{x-1}{4} = \frac{3(x-1)}{12} = \frac{3x-3}{12}\), \(\frac{x+1}{2} = \frac{6(x+1)}{12} = \frac{6x+6}{12}\)

Step 3 - Combine the Numerators

Since all fractions now have the same denominator, we can add the numerators: \(\frac{4x}{12} - \frac{3x-3}{12} + \frac{6x+6}{12}\). Combine the numerators: \(4x - (3x-3) + (6x+6)\)

Step 4 - Simplify the Numerator

Distribute and combine like terms in the numerator: \(4x - 3x + 3 + 6x + 6 = 7x + 9\)

Step 5 - Write the Final Fraction

Put the simplified numerator over the common denominator: \(\frac{7x + 9}{12}\)

Key Concepts

Common Denominators

In algebra, when working with fractions, especially when adding or subtracting, we need a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. This allows us to combine the fractions smoothly.

To find a common denominator, follow these steps:

• Identify the denominators of each fraction. In our exercise, these are 3, 4, and 2.
• Determine the least common multiple (LCM) of these denominators. The LCM of 3, 4, and 2 is 12.
• Rewrite each fraction with this common denominator.

This way, all fractions will be expressed with a denominator of 12, making it easier to add or subtract them later.

Least Common Multiple

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. It's a key concept when finding common denominators.

To find the LCM of a set of numbers:

• List the multiples of each number.
• Identify the smallest multiple that appears in all lists.

For example, with denominators 3, 4, and 2:

• Multiples of 3: 3, 6, 9, 12, 15, ...
• Multiples of 4: 4, 8, 12, 16, 20, ...
• Multiples of 2: 2, 4, 6, 8, 10, 12, ...

Here, 12 is the smallest number that appears in all lists. Therefore, the LCM of 3, 4, and 2 is 12. This LCM is used to rewrite each fraction in our problem with a common denominator of 12.

Fraction Addition in Algebra

Adding fractions in algebra involves more steps than simple arithmetic addition. Here's the process:

• Find a common denominator for all fractions, typically the LCM.
• Rewrite each fraction so they have this common denominator.
• Combine the numerators, keeping the common denominator the same.
• Simplify the resulting fraction, if possible.

For example, if we add \(\frac{x}{3}, \frac{x-1}{4}, \frac{x+1}{2}\), we need to:

• Convert each fraction to have the denominator of 12, resulting in \(\frac{4x}{12}, \frac{3(x-1)}{12}, \frac{6(x+1)}{12}\).
• Combine the numerators: \(\frac{4x}{12} - \frac{3(x-1)}{12} + \frac{6(x+1)}{12}\), which simplifies to \(\frac{4x - 3x + 3 + 6x + 6}{12}\).

This results in the simplified combined fraction: \(\frac{7x+9}{12}\).

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions, including those involving fractions. Like terms are terms that have the same variables raised to the same power. When combining them, just add or subtract their coefficients.

For example, in the expression 4x - (3x - 3) + (6x + 6), we follow these steps:

• Distribute any negative signs: 4x - 3x + 3 + 6x + 6
• Identify and group like terms: 4x, -3x, and 6x are like terms.
• Combine these terms: 4x - 3x + 6x = 7x.
• Add constant terms separately: 3 + 6 = 9.

Putting it all together, we get 7x + 9.

Combining like terms helps make the expression easier to work with, especially when dealing with algebraic fractions.