Problem 10 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(\frac{x-4}{6}-\frac{x-2}{9} \leq \frac{5}{18}\)

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, 13]\).

Step 1: Identify a Common Denominator

The given inequality is \( \frac{x-4}{6}-\frac{x-2}{9} \leq \frac{5}{18} \). The first step is to find a common denominator for the fractions. The denominators are 6, 9, and 18. The least common multiple of these numbers is 18.

Step 2: Rewrite Fractions with the Common Denominator

Convert each term to have the common denominator of 18: \( \frac{x-4}{6} = \frac{3(x-4)}{18} \) and \( \frac{x-2}{9} = \frac{2(x-2)}{18} \). The inequality becomes \( \frac{3(x-4)}{18} - \frac{2(x-2)}{18} \leq \frac{5}{18} \).

Step 3: Combine the Fractions

Since all fractions have the same denominator, combine the numerators: \( \frac{3(x-4) - 2(x-2)}{18} \leq \frac{5}{18} \).

Step 4: Distribute and Simplify the Numerator

Expand and simplify the numerator: \( 3(x-4) = 3x - 12 \) and \( 2(x-2) = 2x - 4 \). This gives us \( 3x - 12 - (2x - 4) \). Simplify this to \( x - 8 \).

Step 5: Simplify the Inequality

Now the inequality is \( \frac{x-8}{18} \leq \frac{5}{18} \). Since the denominators are the same, simplify to \( x-8 \leq 5 \).

Step 6: Solve for x

Add 8 to both sides of the equation: \( x \leq 13 \).

Step 7: Express the Solution in Interval Notation

The solution \( x \leq 13 \) in interval notation is \( (-\infty, 13] \).

Key Concepts

common denominator

When dealing with fractions, particularly when solving inequalities, finding a common denominator is key. The common denominator is the smallest number that each of the denominators can divide into without a remainder. It allows us to transform fractions with different denominators into equivalent fractions with the same denominator. This simplifies calculations and enables easier comparisons between fractions.

In the given inequality, the denominators 6, 9, and 18 require a common denominator so that the fraction terms can be directly compared and combined. We identified the least common multiple (LCM) of 6, 9, and 18, which is 18.

Finding a common denominator often involves:

• Listing the multiples of each denominator.
• Finding the first common multiple from these lists.
• Using this common multiple to rewrite each fraction.

This process ensures that we are working with equivalent values, thus maintaining the balance of the inequality.

interval notation

Once you've solved an inequality, expressing the solution in interval notation provides a clear and concise representation of the solution set. Interval notation uses brackets and parentheses to indicate which numbers are included or excluded in a range.

Here’s how to interpret the different symbols:

• ")" or "(": indicates that the number is not included in the interval.
• "]" or "[": indicates that the number is included in the interval.

For the given inequality, after solving, we determined that the solution is all values of x less than or equal to 13. In interval notation, this is represented as (-∞, 13].

This notation efficiently communicates that any value below 13, going infinitely, is part of the solution, and 13 itself is included in the solution set.

least common multiple

The least common multiple (LCM) is a fundamental concept in algebra, essential for solving problems that involve combining fractions or aligning terms with different denominators. It is the smallest number that is a multiple of two or more numbers.

For the inequality we solved, the LCM of 6, 9, and 18 was crucial. We determined that 18 is the LCM by examining the multiples:

• Multiples of 6: 6, 12, 18, 24...
• Multiples of 9: 9, 18, 27, 36...
• Multiples of 18: 18, 36, 54...

The smallest common number in these lists is 18. Using the LCM allowed us to rewrite the fractions in the inequality so that they all shared the same denominator, simplifying the process of combining and solving them.

Mastering the skill of finding the LCM is vital, as it is frequently used in handling expressions that involve operations with multiple denominators.