Problem 43 - College Algebra with Modeling and Visualization

Question

Solve the inequality graphically. Use set-builder notation. $$ -1 \leq 2 x-1 \leq 3 $$

Step-by-Step Solution

Answer

The solution in set-builder notation is \( \{ x \, | \, 0 \leq x \leq 2 \} \).

Step 1: Understand the Inequality

The inequality to solve is \(-1 \leq 2x - 1 \leq 3\). This represents a compound inequality, meaning we need to find the values of \(x\) that satisfy both parts of the inequality at the same time.

Step 2: Break Down the Compound Inequality

The compound inequality can be split into two separate inequalities: \(-1 \leq 2x - 1\) and \(2x - 1 \leq 3\). We will solve both inequalities separately.

Step 3: Solve the First Inequality

Start with \(-1 \leq 2x - 1\). Add 1 to both sides to get \(-1 + 1 \leq 2x\) or \(0 \leq 2x\). Divide both sides by 2, yielding \(0 \leq x\) or equivalently \(x \geq 0\).

Step 4: Solve the Second Inequality

Now solve \(2x - 1 \leq 3\). Add 1 to both sides to get \(2x \leq 4\). Then, divide by 2 to find \(x \leq 2\).

Step 5: Graph Each Solution on a Number Line

Graph the inequalities \(x \geq 0\) and \(x \leq 2\) on a number line. The points \(x = 0\) and \(x = 2\) are included, so we draw filled circles or dots to represent them. The solution set for the inequality is the intersection of these two, which is the segment between 0 and 2.

Step 6: Write the Solution in Set-Builder Notation

The solution is all values of \(x\) between 0 and 2, inclusive. Thus, in set-builder notation, the solution is \( \{ x \, | \, 0 \leq x \leq 2 \} \).

Key Concepts

Understanding Compound Inequalities

A compound inequality involves two separate inequalities that are combined into one statement. It requires finding values that satisfy both conditions simultaneously. In our problem, the compound inequality is given as \(-1 \leq 2x - 1 \leq 3\). This implies that we are looking for values of \(x\) that make both inequalities true at once.
To tackle a compound inequality, we often split it into two distinct inequalities. For instance, our example splits into

• \(-1 \leq 2x - 1\)
• \(2x - 1 \leq 3\)

The goal is to solve these two inequalities separately and then find the overlapping values that satisfy both conditions.

Using Set-Builder Notation

Set-builder notation is a concise way to express the solution set of an inequality. It describes the set of all elements that satisfy a particular condition. In the notation, we use curly braces "\(\{\)" to enclose the set, and a vertical bar "\(|\)" is used to mean "such that."
For the solution of our compound inequality example, we found that \(x\) can be any number between 0 and 2 inclusive. In set-builder notation, this solution is written as \[ \{ x \, | \, 0 \leq x \leq 2 \} \] which means "the set of all \(x\) such that \(x\) is greater than or equal to 0 and less than or equal to 2."
This format efficiently communicates which numbers are part of the solution without spelling out each number individually.

Techniques for Solving Inequalities

Solving inequalities involves determining the set of values that make the inequality true. The process can be similar to solving regular equations but requires careful attention to the inequality direction.
Here are a few strategies for solving:

• Add or subtract from both sides: This step helps isolate the variable. For instance, in \(-1 \leq 2x - 1\), adding 1 to both sides helps eliminate the constant \(-1\).
• Multiply or divide both sides by a positive number: This step further isolates the variable. With \(0 \leq 2x\), dividing by 2 simplifies the inequality to \(0 \leq x\).

Remember, flipping the inequality sign occurs when you multiply or divide by a negative number, but that doesn't apply to this example since all operations involved positive numbers.
The goal is to simplify until the variable stands alone on one side of the inequality, providing you with the solution range.