Problem 18 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(x \geq 2.1+0.3 x\)

Step-by-Step Solution

Answer

The solution is \([3, \infty)\).

Step 1: Isolate the variable terms

We have the inequality \( x \geq 2.1 + 0.3x \). First, subtract \( 0.3x \) from both sides to get all the terms with \( x \) on the same side: \( x - 0.3x \geq 2.1 \).

Step 2: Simplify the inequality

Combine the terms on the left: \( 1x - 0.3x = 0.7x \). So we have: \( 0.7x \geq 2.1 \).

Step 3: Solve for x

To isolate \( x \), divide both sides of the inequality by 0.7: \( x \geq \frac{2.1}{0.7} \). Simplify the right side: \( \frac{2.1}{0.7} = 3 \). So, we have \( x \geq 3 \).

Step 4: Express the solution set in interval notation

The solution \( x \geq 3 \) can be written in interval notation as \([3, \infty)\).

Key Concepts

Variable Isolation

Variable isolation is a fundamental step in solving inequalities, as it allows us to focus on one variable at a time, making the problem clearer. In algebra, when we talk about isolating the variable, we're essentially moving all the terms containing the variable to one side of the inequality or equation and everything else to the other side.
A crucial part of variable isolation is maintaining balance. What we do to one side, we must do to the other. This is why in the inequality \(x \geq 2.1 + 0.3x\), we subtract \(0.3x\) from both sides. This action moves all \(x\)-related terms to the left.

• Original Inequality: \(x \geq 2.1 + 0.3x\)
• Subtract \(0.3x\) from both sides: \(x - 0.3x \geq 2.1\)
• Simplified to: \(0.7x \geq 2.1\)

This simplification process is key to setting the stage for solving inequalities further.

Interval Notation

Interval notation is a shorthand used in mathematics to express the solution set of inequalities. This form of notation is compact and can represent both finite intervals and infinite sets of values. In our example, where we found \(x \geq 3\), interval notation is particularly useful.
Typically, interval notation uses brackets:

• \([\cdot, \cdot]\): This means the endpoints are included (closed interval).
• \((\cdot, \cdot)\): This means the endpoints are not included (open interval).
• Combinations: \([\cdot, \cdot)\) or \((\cdot, \cdot]\).

In the case of \(x \geq 3\), the interval starts at 3 and stretches to infinity. Infinity always has a parenthesis because it cannot be "reached." Hence, expressed in interval notation, it's written as \([3, \infty)\). This clearly shows all the numbers starting from 3 and moving upwards without end.

Linear Inequalities

Linear inequalities involve expressions in which linear equations or functions are related using inequality signs like \(>\), \(<\), \(\geq\), or \(\leq\). Solving these inequalities involves similar steps to solving linear equations, with one key difference: maintaining the inequality throughout the steps.
Think about linear inequalities as you would linear equations. The same arithmetic operations apply, such as addition, subtraction, multiplication, and division. However, when you multiply or divide the inequality by a negative number, the direction of the inequality sign flips.

• Initial inequality before solving: \(0.7x \geq 2.1\)
• Solving by dividing both sides by 0.7: \(x \geq \frac{2.1}{0.7}\)
• Solution leads to: \(x \geq 3\)

Linear inequalities, just like linear equations, give a solution set, often represented in interval notation. They help describe a range of solutions rather than a single answer, providing more information about conditions under which the inequality holds true.