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Visualizing mathematical solutions on a number line makes complex problems easier to understand. In this concept, you're translating the solution of an inequality into a visual format. Imagine the number line as a ruler where each point represents a real number.
To graph an inequality like \(|x-2|<1\), you'll start by marking specific points that are important based on your solution. In our problem, these points are 1 and 3. Since these are not included in the solution set, we use open circles.
This indicates that the endpoints are not part of the solution. Between these marks, you'll shade the line to show all possible values \(x\) can take, which are between 1 and 3.
The shaded part represents the range of values that satisfy the inequality. Graphing on a number line is a visual cue that enhances understanding and communication of an algebraic solution.

Solving inequalities involves finding the set of values that make an inequality true. It's similar to solving equations, but requires more attention to detail due to the inequality signs.
In our exercise, \(|x-2|<1\), we start by transforming the absolute value inequality into two separate inequalities: \(x-2<1\) and \(2-x<1\).
This allows us to work with simpler linear inequalities. The main trick is remembering that if you multiply or divide an inequality by a negative number, you must flip the inequality sign.
Let's solve \(x-2<1\). Adding 2 to both sides gives \(x<3\). For \(2-x<1\), subtract 2, yielding \(-x<-1\). Dividing by -1, and flipping the sign, we get \(x>1\).
Now you've got a solution that tells you the possible values for \(x\). It's important to check your solution to ensure it satisfies the original inequality.

Understanding algebraic expressions lays the foundation for solving inequalities and equations alike.
Algebraic expressions involve numbers, variables, and operations. In our exercise, the expression \(|x-2|\) indicates the distance between \(x\) and 2 on a number line.
This distance must be smaller than 1, according to \(|x-2| < 1\). When dealing with absolute values, we often split the expression into two cases, as was done earlier with inequalities.
By correctly manipulating these algebraic expressions, you can transform an inequality into something that's manageable.
The goal is always to isolate the variable, which allows us to identify all possible values that satisfy the given condition. Understanding how to navigate and simplify these expressions is key in solving more complex mathematical problems.