Problem 34 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 3 x-1<2(x-3)+1 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -4)\).

Step 1: Expand and Simplify

First, simplify the right side of the inequality. Distribute the 2 across the parentheses:\[2(x-3) = 2x - 6\]Add 1 to this result:\[2x - 6 + 1 = 2x - 5\]Now, the inequality is:\[3x - 1 < 2x - 5\]

Step 2: Isolate the Variable

Subtract \(2x\) from both sides of the inequality to get the variable \(x\) alone on one side:\[3x - 2x - 1 < -5\]This simplifies to:\[x - 1 < -5\]

Step 3: Solve for x

Add 1 to both sides of the inequality to fully isolate \(x\):\[x - 1 + 1 < -5 + 1\]This simplifies to:\[x < -4\]

Step 4: Express the Solution

Since the solution is \(x < -4\), express this in interval notation as:\((-\infty, -4)\)Or, in set-builder notation:\(\{ x \ | \ x < -4 \}\)

Key Concepts

Understanding Set-Builder Notation

Set-builder notation is a way of describing a set by stating a property that its members must satisfy. It is often used in mathematics to specify solutions to equations or inequalities. When you see set-builder notation, it will look like this:

• \( \{ x \ | \ \text{condition} \} \)

The vertical bar "|" is read as "such that," so the expression \( \{ x \ | \ x < -4 \} \) reads as "the set of all \( x \) such that \( x < -4 \)." This notation is useful because it gives a concise way to express complex conditions that describe the set.
Set-builder notation is widely used in algebra and beyond for expressing solutions, reducing the need for lengthy explanations but providing clarity for set members' conditions.

Using Interval Notation

Interval notation is another way of representing sets of numbers, particularly useful for showing ranges of values from inequalities. It condenses the information into a format that is easy to understand and visually intuitive. The format for interval notation will look like:

• The open interval: \((a, b)\), includes all numbers between \(a\) and \(b\) but not \(a\) and \(b\) themselves.
• The closed interval: \([a, b]\), includes \(a\), \(b\), and all numbers in between.
• Open and closed combinations: \((a, b]\) or \([a, b)\), where only one endpoint is included.
• Infinity is always written with open intervals: \((a, \infty)\) or \((−\infty, b)\).

For example, the solution \(x < -4\) is represented as \((-\infty, -4)\) in interval notation, meaning all real numbers less than \(-4\) without including \(-4\) itself. This way of writing makes hierarchies and sequences in sets of numbers easier to interpret and apply.

Mastering Algebraic Manipulation

Algebraic manipulation involves a series of steps used to simplify expressions, solve equations, or solve inequalities like the one seen in the original exercise.
Here's how it works:

• Distribute: Apply the distributive property to eliminate parentheses. For example, distributing \(2(x-3)\) results in \(2x - 6\).
• Combine like terms: Organize terms to simplify. If you have \(-6 + 1\), combine them to get \(-5\).
• Isolate the variable: Rearrange the equation or inequality to get the variable alone on one side. This usually involves adding, subtracting, multiplying, or dividing terms across the equation. In our solution, getting \(x\) alone by subtracting \(2x\) and then adding \(1\) made the inequality simpler.
• Keep the inequality balanced: Any operation on one side must be done to the other side to maintain the inequality's validity.

The step-by-step progression ensures that each operation gets you closer to isolating the variable and finding the solution. Understanding these steps is crucial for tackling more complex algebraic challenges.