Problem 14 - Algebra for College Students

Question

Solve each inequality and graph the solutions. \(|x-2| \geq 1\)

Step-by-Step Solution

Answer

The solution is \(x \leq 1\) or \(x \geq 3\), graphed as two rays with endpoints at 1 and 3.

Step 1: Understand Absolute Inequality

The absolute value inequality \(|x-2| \geq 1\) can be interpreted as the distance between \(x\) and 2 on a number line being at least 1. This means \(x\) can either be greater than or equal to 3, or less than or equal to 1.

Step 2: Set Up Two Separate Inequalities

Since \(|x-2|\) is greater than or equal to 1, this splits into two cases:1. \(x - 2 \geq 1\)2. \(x - 2 \leq -1\).

Step 3: Solve the First Inequality

Solving \(x - 2 \geq 1\), add 2 to both sides to obtain:\[x \geq 3\]

Step 4: Solve the Second Inequality

Solving \(x - 2 \leq -1\), add 2 to both sides to get:\[x \leq 1\]

Step 5: Combine the Solutions

The solutions to the inequalities are \(x \geq 3\) or \(x \leq 1\). This describes a solution where \(x\) is either less than or equal to 1, or greater than or equal to 3.

Step 6: Graph the Solutions

On a number line, shade all values that \(x\) can take. This includes all numbers less than or equal to 1, and all numbers greater than or equal to 3. At 1 and 3, we use solid dots to indicate that those points are included in the solution set.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities help us understand how numbers relate to each other by their distance from a certain point on the number line. Consider \(|x - 2| \geq 1\). The absolute value tells us how far a number is from zero, without considering the direction (positive or negative). In this case, \(x\) is represented as a distance from 2 that is at least 1 unit away. This results in two possible scenarios: either \(x\) is 1 unit or more to the right of 2 (\(x \geq 3\)), or 1 unit or more to the left (\(x \leq 1\)). This kind of inequality can represent an interval or a union of intervals on the number line.

Graphing Solutions

Graphing solutions to inequalities helps us visually see which values of \(x\) satisfy the inequality. After solving an absolute value inequality like \(|x - 2| \geq 1\), we want to present our solutions on a graph. In our case, we identified two critical points, 1 and 3, which mark boundaries. On a number line, you would:

• Shade all numbers to the left of 1, including 1.
• Shade all numbers to the right of 3, including 3.

At the numbers 1 and 3, display solid dots. This highlights that these points are part of the solution and makes it clear that any number less than or equal to 1 or greater than or equal to 3 works for the inequality.

Inequality Problem Solving

Solving inequalities is like solving equations, but we need to look at ranges of values. With absolute value inequalities, the goal is not to find a single number but a range of numbers. Start with splitting the inequality \(|x-2| \geq 1\) into two separate simpler inequalities:

• One \(x - 2 \geq 1\) is straightforward. Adding 2 to both sides gives \(x \geq 3\).
• The other, \(x - 2 \leq -1\), also simplifies by adding 2, resulting in \(x \leq 1\).

By solving each inequality independently, you identify the comprehensive range of solutions needed to satisfy the original absolute value inequality. This step-by-step approach ensures accuracy and clarity, setting a solid foundation for tackling more complex inequalities in the future.

Number Line Solutions

A number line is a useful tool for representing solutions of inequalities, especially when dealing with absolute values. To represent the solution for \(|x - 2| \geq 1\), you draw a horizontal line with numbers marked according to context:

• Identify key numbers from solutions, here 1 and 3.
• Shade the entire segment left of 1, and extend indefinitely to the left.
• Then, start from number 3, shade all the way right towards infinity.

The shading tells us all possible solutions for \(x\) that satisfy the inequality. Use solid dots on 1 and 3 as these values are included in the solution. This method gives a clear and visual representation of how solutions work for inequalities.