Problem 16 - College Algebra

Question

Solve the inequality involving absolute value. Write your final answer in interval notation. $$ |2 x+3|<7 $$

Step-by-Step Solution

Answer

The solution is \((-5, 2)\).

Step 1: Understand Absolute Value Inequality

The given inequality is \(|2x+3|<7\). Absolute value inequalities like this one tell you that the expression inside the absolute value is between a range of two numbers. Specifically, the inequality \(|A| < B\) is equivalent to \(-B < A < B\).

Step 2: Set Up the Compound Inequality

Based on the rule from Step 1, rewrite the inequality \(|2x+3|<7\) as two separate inequalities without the absolute value: \(-7 < 2x + 3 < 7\).

Step 3: Solve the Left Inequality

Solve the first part of the compound inequality: \(-7 < 2x + 3\).Subtract 3 from both sides:\(-7 - 3 < 2x\) which simplifies to \(-10 < 2x\).Then, divide both sides by 2 to isolate \(x\):\(-5 < x\).

Step 4: Solve the Right Inequality

Solve the second part of the compound inequality: \(2x + 3 < 7\).Subtract 3 from both sides:\(2x < 7 - 3\) which simplifies to \(2x < 4\).Then, divide both sides by 2 to isolate \(x\):\(x < 2\).

Step 5: Combine the Results

The solution to the compound inequality \(-7 < 2x + 3 < 7\) is a combination of the solutions from Steps 3 and 4. Combine these results to form the inequality \(-5 < x < 2\).

Step 6: Write the Solution in Interval Notation

The inequality \(-5 < x < 2\) can be expressed in interval notation. Since both \(-5\) and \(2\) are NOT included (indicated by the \(<\) signs), we use parentheses:The solution in interval notation is \((-5, 2)\).

Key Concepts

Interval Notation

Interval notation is a shorthand way of expressing ranges of numbers in mathematics. It is especially useful for expressing the solution sets of inequalities. Interval notation uses parentheses \((\) and \()\) to indicate that endpoints are not included, and brackets \([\) and \()]\) to show that endpoints are included. For example, the range of numbers greater than -5 and less than 2 is written as \((-5, 2)\), meaning -5 and 2 are not part of the solution. If the endpoints were included, the notation would change to \([-5, 2]\).

When solving inequalities, especially absolute value inequalities, translating your final solution into interval notation provides a clear and concise representation. It immediately tells you the span of values that satisfy the inequality. In the problem discussed, the solution \((-5, 2)\) implies that all numbers from just above -5 to just below 2 satisfy \(|2x+3|<7\). To read interval notation correctly:

• Use parentheses for a "less than" or "greater than" condition.
• Use brackets for "less than or equal to" or "greater than or equal to" conditions.

This method keeps your expressions tidy and easy to interpret, which is why interval notation is so widely used.

Compound Inequalities

A compound inequality involves two separate inequalities that are joined together. This is common in absolute value inequalities and is fundamental to understanding how we solve them. In the original exercise, the inequality \(|2x+3| < 7\) means that the expression within the absolute value sign is bounded by -7 and 7. This is written as two separate inequalities: \(-7 < 2x+3 < 7\).

Compound inequalities can be either "and" or "or" types:

• "And" compound inequalities mean both conditions must be true simultaneously. This is the case in our example: both \(-7 < 2x+3\) and \(2x+3 < 7\) need to be true to satisfy \(|2x+3| < 7\).
• "Or" compound inequalities imply that at least one of the conditions must be true. This is typical in absolute value inequalities using the form \(|A| > B\).

By solving each part separately, you narrow down the possible values of the variable \(x\) that satisfy the original inequality. These values are then combined to give the range of valid solutions, which is ultimately expressed using interval notation, like \((-5, 2)\) in our example.

Algebraic Solutions

Solving inequalities algebraically involves a series of steps focused on logically isolating the variable. In the context of absolute value inequalities, this process also involves rewriting the inequality as a compound inequality. The practice of solving each part separately ensures that you carefully consider the constraints imposed by the inequality.

For the inequality \(|2x+3|<7\), the process is as follows:

• Rewrite the inequality without the absolute value, such as \(-7 < 2x+3 < 7\), forming a compound inequality.
• Address each inequality separately:
  - Solve \(-7 < 2x+3\) to find \(-5 < x\).
  - Solve \(2x+3 < 7\) to find \(x < 2\).
• Combine these individual results to identify the range of values for \(x\): \(-5 < x < 2\).

Algebraic solutions require careful attention to positive and negative numbers, especially when dividing both sides of an inequality by negative numbers—something not present in this problem but crucial to remember in others. Once you solve the inequalities, you should translate your solution into interval notation. This ensures clarity and precision, presenting your answer in a universally recognized format.