Problem 14 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(0.08 x+0.09(2 x) \geq 130\)

Step-by-Step Solution

Answer

The solution in interval notation is \([500, \infty)\).

Step 1: Distribute and Combine Like Terms

First, simplify the inequality by distributing the 0.09 across the terms inside the parentheses. This gives us:\[0.08x + 0.18x \geq 130\]Next, combine the like terms on the left side:\[0.26x \geq 130\].

Step 2: Solve for x

To isolate \(x\), divide both sides of the inequality by 0.26:\[x \geq \frac{130}{0.26}\].Performing the division gives:\[x \geq 500\].

Step 3: Express the Solution Set in Interval Notation

Since the inequality sign is \(\geq\), the solution includes 500 and all numbers greater than 500. Thus, the solution set in interval notation is:\[[500, \infty)\].

Key Concepts

Interval Notation

Interval notation is a very efficient way to represent a range of numbers along the number line. It's compact and easy to understand. In interval notation, we use brackets and parentheses:

• A square bracket "[" or "]" indicates that an endpoint is included in the interval. We call this "inclusive" notation.
• A parenthesis "(" or ")" shows that an endpoint is not included, known as "exclusive" notation.

For example, the interval \([500, \infty)\) includes the number 500 and all numbers greater than 500, but not infinity itself, because infinity is a concept rather than a finite number.
Learning to express solution sets in interval notation helps to succinctly communicate which numbers satisfy an inequality.

Distributive Property

The distributive property is key to simplifying expressions involving parentheses. It allows you to multiply a single term by each term inside a parenthesis. In mathematical terms, for any numbers \( a \), \( b \), and \( c \), the property states:\[a(b+c) = ab + ac\]In the inequality \(0.08 x+0.09(2 x) \geq 130\), the distributive property helps us simplify:\(0.09(2x)\).
By distributing, we get \[0.09 \times 2x = 0.18x\].
Using the distributive property allows us to break down complex problems into simpler parts, making it a powerful tool in algebra.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms with the same variable part to simplify an expression. Like terms in an algebraic expression have exactly the same variables and exponents, though their coefficients can differ.
In our example:\(0.08x + 0.18x\), both terms feature the variable \(x\).
To combine them, we simply add their coefficients:\(0.08 + 0.18 = 0.26\). The simplified expression becomes:\(0.26x\).
By combining like terms, we effectively streamline the expression, making it easier to solve.

Solving Linear Inequalities

Solving linear inequalities involves finding which values of the variable satisfy the inequality condition. When solving the inequality:\(0.26x \geq 130\), we aim to isolate \(x\) on one side.

• The first step is to divide both sides by 0.26, yielding the expression: \(x \geq \frac{130}{0.26}\).
• Perform the division to get a numerical result: \(x \geq 500\).

It's crucial to remember that if you ever multiply or divide by a negative number during the process, you must flip the inequality sign.
However, in this problem, multiplication and division are positive, so the inequality sign stays \(\geq\). Solving inequalities like this is similar to solving equations but requires careful attention to the inequality sign.