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Chapter 2: Problem 111

Solve. Graph the solutions on a number line and give the corresponding interval notation. $$|2 x+3|<1$$

Short Answer

Expert verified
The solution is \(-2 < x < -1\), or \((-2, -1)\) in interval notation.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality \(|2x+3| < 1\) means that the expression inside the absolute value, \(2x+3\), is between -1 and 1. Therefore, we can write two separate inequalities: \(-1 < 2x + 3 < 1\).
02

Solve the Left Part of the Inequality

First, solve \(-1 < 2x + 3\). Subtract 3 from both sides to get \(-4 < 2x\). Then divide by 2 to obtain \(-2 < x\).
03

Solve the Right Part of the Inequality

Next, solve \(2x + 3 < 1\). Subtract 3 from both sides to get \(2x < -2\). Then divide by 2 to obtain \(x < -1\).
04

Combine the Inequalities

Combine the results from the two inequalities to get the solution: \(-2 < x < -1\). This is the range of values for \(x\) that satisfy the original inequality.
05

Graph the Solution on a Number Line

Draw a number line and shade the region between \(x = -2\) and \(x = -1\). Use open circles at -2 and -1 to indicate that these endpoints are not included in the solution set.
06

Write the Interval Notation

The solution, \(-2 < x < -1\), can be written in interval notation as \((-2, -1)\). This notation indicates the continuous range of x-values between -2 and -1, excluding -2 and -1 themselves.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

number line graphing
Graphing on a number line is a visual way to represent inequalities. It helps in understanding which values are included in the solution set. For the inequality \(-2 < x < -1\), we only include values between -2 and -1. This interval is shown with open circles on the number line, indicating that -2 and -1 are not part of the solution.

To graph it:
  • Draw a straight line and label a section of it with numbers, say from -3 to 0.
  • Locate -2 and -1 on this line.
  • Place open circles at -2 and -1 because these numbers are not included in the solution set (use of open circles for values not included is important).
  • Shade the space between these open circles to show all the possible values of \(x\).
The number line makes it easy to see which sections satisfy the inequality.
interval notation
Interval notation is a concise way to express the range of solutions in inequalities. For our solution \(-2 < x < -1\), interval notation captures this as \((-2, -1)\). The parentheses \(()\) indicate that -2 and -1 are not part of the solution, aligning with the open circles on the number line graph.

Here's how interval notation works:
  • Parentheses \(()\) are used for numbers not included in the solution set.
  • Brackets \([]\) mean the number is included in the set.
  • The format is always \((\text{lower bound}, \text{upper bound})\) or \([\text{lower bound}, \text{upper bound}]\).
For instance, if you had \(x \geq -2\) or \(x \leq -1\), the interval would change to \([-2, -1]\) if both ends are included. It provides a compact and precise expression of the solution set.
solving compound inequalities
Compound inequalities involve two simple inequalities combined. The solution must satisfy both to be true. After breaking down the absolute value inequality \(|2x+3| < 1\), we split it into two: \(-1 < 2x+3 < 1\). This creates a compound inequality requiring careful solving.

Steps:
  • Write the compound inequality, like \(-1 < 2x+3 < 1\).
  • Solve the left side: subtract 3 from \(-1\) giving \(-4 < 2x\), then divide by 2 to find \(x > -2\).
  • Solve the right side: subtract 3 from \(1\) to get \(2x < -2\), then divide by 2 to find \(x < -1\).
  • Combine both to yield \(-2 < x < -1\), meaning both conditions must be met simultaneously.
Understanding compound inequalities is about seeing how two ranges overlap to form a complete solution. It demonstrates how individual conditions interact in absolute value scenarios.

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