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When solving an absolute inequality, we often need to find a range of values that satisfy the given condition. In the case of \(|10x + 5| < 25\), we treated the inequality as two separate inequalities. One being \(10x + 5 < 25\) and the other \(10x + 5 > -25\). By solving these inequalities, we find the values for \(x\) that make the original statement true.

You should solve each inequality step by step. For the first, subtract 5 and then divide by 10, leading to \(x < 2\). For the second, subtract 5, then divide by 10 to get \(x > -3\). The solution set is the intersection of these results, which gives us \(-3 < x < 2\).

This means \(x\) is any number that lies between -3 and 2, without including -3 and 2 themselves. Understanding this intersection helps us find the most precise solution within the real number line.

Interval notation is a simplified way to express a range of numbers. It's often used in completing and presenting solutions to inequalities. In our problem, we have found the solution set to be \(-3 < x < 2\).

Let's express this using interval notation, which uses parentheses and brackets to show open and closed ends:

• If the interval is open, meaning the endpoints are not included, use parentheses: \( (a, b) \).
• If the interval is closed, meaning the endpoints are included, use brackets: \( [a, b] \).

Therefore, the solution set of \(-3 < x < 2\) is expressed in interval notation as \((-3, 2)\).

This notation is concise and makes it easy to quickly identify which numbers are included in the solution set.

Graphing inequalities on a number line offers a visual representation of the solution set, making it easier to understand the range of values. For \(-3 < x < 2\), we use an open circle to indicate that -3 and 2 are not included in the solution.

Here's how to graph it step-by-step:

• Draw a number line.
• Place open circles at points -3 and 2 to show these endpoints are not part of the solution.
• Shade the region between the open circles. This shaded part represents all the values of \(x\) that satisfy \(-3 < x < 2\).

This visualization helps confirm that any number within the shaded region is a part of the solution to the inequality \(|10x + 5| < 25\). It's a powerful tool to intuitively grasp concepts and check for errors in solving inequalities.