Problem 33 - Algebra 1: Concepts and Skills

Question

Write the difference in simplest form. $$ \frac{2 x-1}{3 x}-\frac{1}{11} $$

Step-by-Step Solution

Answer

The difference in simplest form is \( \frac{19x-11}{33x} \).

Step 1: Identify the Common Denominator

The denominators in this case are \(3x\) and \(11\). Their least common denominator (LCD) is obtained by multiplying these two together, which gives \(33x\).

Step 2: Convert each fraction into an equivalent fraction with this common denominator

Consequently, we recreate each fraction as an equivalent fraction with denominator \(33x\):\n\( \frac{2x-1}{3x} * \frac{11}{11}= \frac{11 * (2x-1)}{33x} \)\n\( \frac{1}{11} * \frac{3x}{3x}= \frac{3x}{33x} \)

Step 3: Subtract the two fractions

We can now subtract the two fractions since they have a common denominator: \( \frac{11 * (2x-1)}{33x} - \frac{3x}{33x} \)

Step 4: Simplify the result

Simplify the result to obtain the final answer: \( \frac{22x - 11 - 3x}{33x} = \frac{19x-11}{33x} \)

Key Concepts

Least Common Denominator

Understanding how to find the Least Common Denominator (LCD) is key when working with rational expressions. The LCD is the smallest expression that both denominators can divide into without leaving a remainder. Here's how to find it simply:

• First, identify the different denominators. In this exercise, the denominators are \(3x\) and \(11\).
• Next, determine the products you need to reach a common denominator. In this case, multiplying \(3x\) by \(11\) ensures both original denominators can transform into \(33x\), giving us our LCD.

Finding the LCD helps facilitate operations like addition or subtraction of fractions by simplifying them to have the same base.

Fraction Subtraction

Subtracting fractions might seem tricky at first, but it's all about making sure they share the same denominator and then subtracting the numerators. Here’s a simple walk-through:

• Convert each fraction to an equivalent fraction with the LCD. We transformed \(\frac{2x-1}{3x}\) into \(\frac{11(2x-1)}{33x}\) and \(\frac{1}{11}\) into \(\frac{3x}{33x}\).
• Now, with these equal denominators, you only need to subtract the numerators: this means doing \(11(2x-1) - 3x\).

Once this subtraction is complete, the expression is further simplified to make the solution more straightforward.

Simplifying Expressions

The final step in this problem is simplifying the expression you got from subtracting the fractions. Simplification involves reducing the expression to its most basic form:

• After subtracting, we have the result \(\frac{22x - 11 - 3x}{33x}\).
• Combine like terms in the numerator: \(22x - 3x = 19x\).
• So, the expression becomes \(\frac{19x - 11}{33x}\).

This is the simplest form of the expression, meaning that no further reduction is possible. Simplifying expressions helps in understanding and interpreting mathematical results efficiently.