Problem 49 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(-2<-b+3<5\)

Step-by-Step Solution

Answer

The solution is \(-2 < b < 5\) or \((-2, 5)\).

Step 1: Simplify the Left Inequality

We begin solving the compound inequality by breaking it into two separate inequalities. The left inequality is \(-2 < -b + 3\). First, subtract 3 from both sides to isolate \(-b\) on the right side:\[-2 - 3 < -b\]Simplifying gives us:\[-5 < -b\]

Step 2: Solve the Left Inequality

Now, we solve for \(b\) in the inequality \(-5 < -b\). First, multiply or divide each term by -1, which flips the inequality sign:\[5 > b\]Or equivalently, \(b < 5\).

Step 3: Simplify the Right Inequality

Next, solve the right part of the compound inequality, \(-b + 3 < 5\). Subtract 3 from both sides:\[-b < 5 - 3\]Simplifying gives us:\[-b < 2\]

Step 4: Solve the Right Inequality

To solve \(-b < 2\), multiply or divide each term by -1 (flipping the inequality sign):\[b > -2\]

Step 5: Combine the Inequalities

We now have two inequalities from Steps 2 and 4: \(b > -2\) and \(b < 5\). Combining them gives the compound inequality:\(-2 < b < 5\)

Step 6: Write the Solution in Interval Notation

The solution to the compound inequality \(-2 < b < 5\) in interval notation is written as:\((-2, 5)\)This denotes all numbers between -2 and 5, not including -2 and 5.

Key Concepts

Inequality Solutions

When dealing with compound inequalities like \(-2 < -b + 3 < 5\), we must dissect them into separate inequalities that are easier to manage. A compound inequality contains multiple simple inequalities that are usually connected by 'and' or 'or'. Here, we break down the problem by dealing with each part independently.

• First, simplify each part by isolating the variable. In this example, we solve both \(-2 < -b + 3\) and \(-b + 3 < 5\) separately.
• The key operation with inequalities is sensitive to multiplying or dividing by negative numbers. This action requires flipping the inequality sign to maintain the logic of the inequality.

Logical consistency is crucial when handling inequalities; each step requires precise operations to derive a true solution. Ultimately, solving inequalities is about finding all values of the variable that will make the compound statement true.

Interval Notation

Interval notation provides a shorthand method for expressing ranges of values. It's an elegant way to depict solutions of inequalities. In the case of the inequality solutions \(-2 < b < 5\), we convert this into interval notation by framing it as \((-2, 5)\).

• The parentheses \(()\) are used when the boundaries are not included in the solution set, indicating that \(b\) does not equal \(-2\) or \(5\).
• Square brackets \([]\) would be employed if the endpoints were part of the solution (e.g., \[a, b\]).

This concise form efficiently conveys all possible values of \(b\) that satisfy the compound inequality. Learning to read and write interval notation is essential for clearly communicating the results of your solutions.

Graphing Inequalities

Graphing is a visual representation that complements algebraic solutions, offering a clearer insight into the solution set. To graph the compound inequality \(-2 < b < 5\), follow these steps:

• Start by drawing a number line.
• Indicate the values \(-2\) and \(5\) on this line. Use open circles at these points to show that they are not included in the solution.
• Shade the area between \(-2\) and \(5\) on the number line. This shading signifies that all numbers between these points are part of the solution.

The graph provides a quick, clear visual that supports the algebraic solution, enabling you to easily see which values \(b\) can take. Graphs are handy tools for double-checking your work and confirming your understanding of inequalities.