Problem 59 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|x-1| \leq 2$$

Step-by-Step Solution

Answer

The solution set to the inequality \(|x-1| \leq 2\) is \([-1, 3]\)

Step 1: Understand the Absolute Value

Recall that the absolute value of \(x-1\) is simply the distance between \(x\) and 1 on the number line. Given that \(|x-1| \leq 2\), this indicates that the distance of \(x\) from 1 should be less than or equal to 2.

Step 2: Break Down the Inequality

Since the distance could be less than or equal to 2 above or below 1, this provides two inequalities: \(x-1 \leq 2\) and \(-(x-1) \leq 2\). Which gives us \(x \leq 3\) and \(-x+1 \leq 2\) respectively. Then, \(-x+1 \leq 2\) further simplifies to \(x \geq -1\) when both sides are multiplied by -1.

Step 3: Graph the Solution Set

The solution to this inequality will be all \(x\) that satisfy both \(x \leq 3\) and \(x \geq -1\). This would be all values of \(x\) that fall in the interval between -1 and 3 on the number line, inclusive.

Step 4: Write the Solution in Interval Notation

The solution set is expressed in interval notation as \([-1, 3]\). This means that it includes all real numbers \(x\) such that -1 \(\leq\) \(x\) \(\leq\) 3. Both endpoints of this interval are closed (included in the interval).

Key Concepts

Understanding Inequality Solutions

When it comes to solving inequalities with absolute value, it's crucial to understand that we are considering the distance from a specific point. For our example, \(|x-1| \leq 2\), we are looking at the distance of \(x\) from 1 being at most 2 units away on the number line. This inequality is expressing two conditions: one when \(x\) is greater than or equal to 1 and the other when \(x\) is less than 1.

The two resulting inequalities \(x \leq 3\) and \(x \geq -1\) represent the range of x-values that satisfy the original condition. These inequalities help us form a mental picture of a segment on the number line where all the values of \(x\) that solve the inequality reside. The act of solving these inequalities involves combining the possible values of \(x\) into one coherent solution.

Interval Notation Explained

In mathematics, interval notation is a method used to describe the set of numbers lying between two endpoints. The notation includes square brackets or parentheses to tell us whether these endpoints are included or excluded, respectively. In the case of our derived solution \(x \leq 3\) and \(x \geq -1\), the interval notation is \[ -1, 3 \] which signifies that the endpoints -1 and 3 are part of the solution.

A square bracket, \[ or \], indicates that the endpoint is included, denoting a 'closed interval'. Conversely, a parenthesis, \( or \)), would suggest that the endpoint is not included, meaning an 'open interval'. For example, if our solution were \(x < 3\) and \(x > -1\), the interval notation would be \( -1, 3 \) to reflect that -1 and 3 are not solutions themselves.

Number Line Graphing

Graphing on a number line offers a visual representation of all the possible solutions to an inequality. For \(x \leq 3\) and \(x \geq -1\), imagine a horizontal line labeled with numbers. You'll then shade the segment that lies between -1 and 3 to indicate that every point in this shaded area is a solution to the inequality.

For closed intervals, dot the endpoints to signify that they are included in the solution set. If they were not included, as in an open interval, the dots would be left open or hollow. Graphical representation is a helpful tool to pair with interval notation since it adds clarity and a helpful visual cue for those who prefer to process information graphically.