Problem 1 - Intermediate Algebra

Question

For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. $$ \frac{2}{5} x+\frac{1}{3} x>\frac{44}{15} $$

Step-by-Step Solution

Answer

The solution set in interval notation is \((4, \infty)\).

Step 1: Combine Like Terms

First, combine the like terms on the left side of the inequality. Add \( \frac{2}{5}x \) and \( \frac{1}{3}x \) by finding a common denominator. The common denominator for 5 and 3 is 15. Rewrite each fraction with this common denominator: \( \frac{2}{5} = \frac{6}{15} \) and \( \frac{1}{3} = \frac{5}{15} \). Thus, \( \frac{2}{5}x + \frac{1}{3}x = \frac{6}{15}x + \frac{5}{15}x = \frac{11}{15}x \).

Step 2: Isolate the Variable

Now, divide both sides of the inequality by \( \frac{11}{15} \) to isolate the variable. The inequality becomes \( x > \frac{44}{15} \div \frac{11}{15} \).

Step 3: Simplify the Right Side

When dividing fractions, you multiply by the reciprocal. Thus, \( \frac{44}{15} \div \frac{11}{15} = \frac{44}{15} \times \frac{15}{11} = \frac{44}{11} = 4 \). So the inequality \( x > 4 \).

Step 4: Express Solution in Interval Notation

With the inequality \( x > 4 \), express the solution in interval notation. Since \( x \) is greater than 4 but not equal to 4, the interval notation is \((4, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is used to express the solution set of an inequality in a concise format. When using interval notation to describe the set of values that solve the given inequality, we make use of parentheses

• Parentheses, \((\underline{\phantom{xxx}}\underline{\phantom{xxx}})\), indicate that an endpoint is not included in the set.
• Brackets, \([\underline{\phantom{xxx}}\underline{\phantom{xxx}}]\), indicate that an endpoint is included in the set.

In our solved inequality, \(x > 4\), you note that the number 4 is not part of the solution set—all numbers greater than 4 are included instead. Thus, 4 is not included, and the interval notation becomes \((4, \infty)\).

When using \(\infty\) in interval notation, always use a parenthesis since infinity itself is a concept rather than a fixed number and is never included in the set.

Combining Like Terms

Combining like terms is a fundamental concept in algebra that helps simplify expressions and solve equations or inequalities more easily. Like terms are terms whose variables (and their exponents) are the same. In expressions involving fractions, like the one in the exercise, you often need to find a common denominator before you can combine them successfully.

Here's a recap of the steps needed to combine like terms in the exercise

• Identify the terms involving the same variable, such as \(\frac{2}{5}x\) and \(\frac{1}{3}x\).
• Find a common denominator—in the case of fractions to add them up. For \(\frac{2}{5}\) and \(\frac{1}{3}\), the common denominator is 15.
• Rewrite each fraction with the common denominator: \(\frac{6}{15}x\) plus \(\frac{5}{15}x\).
• Add the numerators: \(\frac{6x+5x}{15} = \frac{11x}{15}\).

Combining like terms is essential to keep equations neat and manageable, helping you prepare them for the vital next steps like isolating variables.

Isolating Variables

Isolating the variable is a key step in solving equations and inequalities. The goal is to have the variable by itself on one side of the equation or inequality to easily determine its value or range.

• In this problem, the expression \(\frac{11}{15}x > \frac{44}{15}\) was simplified down by combining like terms.
• Next, to isolate \(x\), divide both sides of the inequality by the coefficient of \(x\), which is \(\frac{11}{15}\).
• Dividing by a fraction means multiplying by the reciprocal of that fraction. Therefore, \(\frac{44}{15} \div \frac{11}{15}\) becomes \(\frac{44}{15} \times \frac{15}{11} = 4\).
• By performing these arithmetic operations, you isolate \(x\) and get \(x > 4\).

Successfully isolating variables turns complex expressions into simpler solutions, allowing you to clearly express the solution in interval notation or any required form.