Problem 9 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(\frac{x+3}{8}-\frac{x+5}{5} \geq \frac{3}{10}\)

Step-by-Step Solution

Answer

The solution is \( (-\infty, -\frac{37}{3}] \).

Step 1: Set up the inequality equation

The inequality given is \( \frac{x+3}{8} - \frac{x+5}{5} \geq \frac{3}{10} \). Our goal is to solve for \(x\) and express the solution in interval notation.

Step 2: Find a common denominator

The fractions involved have denominators 8, 5, and 10. To combine them, we need a common denominator. The least common multiple of 8, 5, and 10 is 40. We will rewrite each fraction with 40 as the denominator.

Step 3: Convert fractions to have a denominator of 40

Convert \(\frac{x+3}{8} \) to \( \frac{5(x+3)}{40} \), convert \(\frac{x+5}{5} \) to \( \frac{8(x+5)}{40} \), and convert \(\frac{3}{10} \) to \( \frac{12}{40} \).

Step 4: Substitute the converted fractions back into the inequality

The inequality is rewritten as \( \frac{5(x+3)}{40} - \frac{8(x+5)}{40} \geq \frac{12}{40} \).

Step 5: Simplify the expression

Combine the fractions on the left side: \(\frac{5(x+3) - 8(x+5)}{40} \geq \frac{12}{40} \).

Step 6: Distribute and combine like terms

Simplify the expression by distributing: \(5x + 15 - 8x - 40\), which simplifies to \(-3x - 25\).

Step 7: Remove the fraction by multiplying by 40

Multiply both sides by 40 to get: \(-3x - 25 \geq 12\).

Step 8: Solve for x

Add 25 to both sides: \(-3x \geq 37\), and then divide by -3 to solve for \(x\). Remember, dividing by a negative reverses the inequality sign, resulting in \(x \leq -\frac{37}{3}\).

Step 9: Convert to interval notation

The solution \(x \leq -\frac{37}{3}\) is represented in interval notation as \( (-\infty, -\frac{37}{3}] \).

Key Concepts

Solution Sets

In algebra, solution sets are the collection of all possible solutions that satisfy a given equation or inequality. For example, in inequalities, we're looking for all values of the variable that make the inequality true. In the given inequality \( \frac{x+3}{8} - \frac{x+5}{5} \geq \frac{3}{10} \), our aim is to find all possible values of \(x\) that satisfy this condition.This involves a series of steps:

• Rearranging the inequality to a form that allows us to solve for \(x\).
• Finding common denominators to combine fractions.
• Simplifying the expressions to isolate the variable \(x\) on one side.

Once all these are performed, the solution to the inequality \(-3x - 25 \geq 12\) gives \(x \leq -\frac{37}{3}\). This tells us that the set of all \(x\) which are less than or equal to \(-\frac{37}{3}\) forms the solution set for this inequality. These values are represented in interval notation to give a clear range of the solution.

Interval Notation

Interval notation is a way of representing the set of solutions in a precise manner. It uses parentheses \(()\) and brackets \([]\) to show which numbers are included in or excluded from the solution set.For the inequality \(x \leq -\frac{37}{3}\):

• The bracket \([\) is used because \(-\frac{37}{3}\) is included in the solution set (meaning it satisfies the inequality).
• The parenthesis \(()\) is used with \(-\infty\) because infinity is not a specific number and cannot be included.

Thus, the interval notation for the solution is \((-\infty, -\frac{37}{3}]\), indicating that \(x\) can take any value less than or equal to \(-\frac{37}{3}\). This concise form clearly communicates the range of solutions without listing every possible value.

Common Denominator

A common denominator is essential when dealing with fractional expressions in algebra. It allows for the combination of fractions when adding, subtracting, or comparing them. The common denominator is a shared multiple of all individual denominators.In the inequality \( \frac{x+3}{8} - \frac{x+5}{5} \geq \frac{3}{10} \), the denominators are 8, 5, and 10. The least common multiple of these numbers is 40, which becomes our common denominator.This conversion is done by multiplying the numerator and the denominator of each fraction by a number that makes the denominator equal to the common denominator:

• The fraction \(\frac{x+3}{8}\) is converted to \(\frac{5(x+3)}{40}\).
• The fraction \(\frac{x+5}{5}\) is converted to \(\frac{8(x+5)}{40}\).
• The fraction \(\frac{3}{10}\) is converted to \(\frac{12}{40}\).

Using a common denominator simplifies the process of solving inequalities with fractions as it allows for straightforward arithmetic operations on the fractions involved.