Problem 59 - Elementary and 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

Solution: \( b \in (-2, 5) \). Graph between \(-2\) and \(5\) with open circles.

Step 1: Isolate the variable part on both inequalities

Start by isolating the variable \( -b+3 \) in both parts of the compound inequality \(-2 < -b+3 < 5\). To do this, subtract 3 from all three parts to have a new inequality: \(-2 - 3 < -b+3 - 3 < 5 - 3\). This simplifies to \(-5 < -b < 2\).

Step 2: Solve for the positive variable

After isolating \(-b\), we need to solve for \(b\). We can multiply the entire inequality by \(-1\), which will reverse the inequality signs, resulting in \(5 > b > -2\). This can also be written as \(-2 < b < 5\).

Step 3: Express the solution in interval notation

The solution \(-2 < b < 5\) can be written in interval notation as \((-2, 5)\), which means \(b\) is greater than \(-2\) and less than \(5\).

Step 4: Graph the solution set

To graph the solution set, draw a number line. Mark open circles at \(-2\) and \(5\) since \(b\) does not include \(-2\) and \(5\) themselves. Shade the region between \(-2\) and \(5\) to indicate all values \(b\) can take.

Key Concepts

Interval Notation

Interval notation provides a simple and concise way to represent ranges of numbers in algebra. When we solved the compound inequality \(-2 < b < 5\), the solution was expressed using interval notation as \((-2, 5)\). The parentheses \(()\) indicate that both endpoints \(-2\) and \(5\) are not included in the set, known as "open intervals."

• If an endpoint is included, we would use square brackets \([ ]\), indicating a "closed interval."
• For example, if our solution was \(-2 \leq b < 5\), it would be written as \([-2, 5)\).

Expressing solutions in this form helps communicate the range of values a variable can take in a straightforward manner. Make sure to pay attention to whether the inequality uses "less than" or "less than or equal to," as this determines whether brackets or parentheses are used.

Number Line Graph

Graphing compound inequalities on a number line offers a visual way to interpret your solution. For the compound inequality \(-2 < b < 5\), we start by drawing a simple horizontal line.

• Next, mark the key points on the line, which are the numbers \(-2\) and \(5\).
• Use open circles for these points to show they are not included in the solution.
• Shade the region between these two points to represent the range of solutions for \(b\).

This shaded region helps you quickly see that values between \(-2\) and \(5\) satisfy the inequality, while any value outside this range does not. Remember, the number line graph complements the algebraic solution, giving a more intuitive grasp of the problem at hand.

Isolating Variable

Isolating the variable is a foundational step in solving inequalities and equations. The goal is to have the variable on one side of the inequality to better understand what values it can take. Let's walk through the process again for our compound inequality \(-2 < -b+3 < 5\).

• Start by performing inverse operations to eliminate any terms around the variable. Here, subtract \(3\) from all parts of the inequality: \(-2 - 3 < -b < 5 - 3\), simplifying to \(-5 < -b < 2\).
• Since we want \(b\) alone, and it's currently negative, multiply through by \(-1\) (this reverses the inequality signs) to isolate \(b\): \(5 > b > -2\).
• You can rewrite it in the more common left-to-right order, \(-2 < b < 5\).

By carefully isolating the variable, we ensure we accurately determine the solution set. Remember, reversing inequalities when multiplying or dividing by a negative number is crucial in maintaining the correct expression of the solution.