Problem 49 - 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

The solution is \((-2, 5)\), and the graph is a line segment between -2 and 5 on the number line.

Step 1: Understand the Compound Inequality

The given compound inequality is \(-2 < -b + 3 < 5\). This means that \(-b + 3\) must be greater than \(-2\) and less than \(5\), simultaneously.

Step 2: Solve the First Inequality Part

We start by solving \(-2 < -b + 3\). First, subtract 3 from both sides to isolate the term with \(b\): \[-2 - 3 < -b\] \[-5 < -b\] Now, multiply or divide by \(-1\) to solve for \(b\), remembering to flip the sign: \[5 > b\] or equivalently \(b < 5\).

Step 3: Solve the Second Inequality Part

Next, solve \(-b + 3 < 5\). Subtract 3 from both sides: \[-b < 2\] Now, multiply or divide by \(-1\) to solve for \(b\), remembering to flip the sign: \[b > -2\].

Step 4: Combine the Solutions

Now combine the two inequalities \(b < 5\) and \(b > -2\). The compound inequality for \(b\) is \(-2 < b < 5\).

Step 5: Represent the Solution on a Number Line and Interval Notation

The solution \(-2 < b < 5\) is represented on a number line as a segment between -2 and 5, not including the endpoints. In interval notation, this is written as \((-2, 5)\).

Key Concepts

Interval Notation

Interval notation is a shorthand way to describe a set of numbers that fall within a specific range on the number line. It's very handy when dealing with inequalities. The compound inequality \(-2 < b < 5\) means the values of \(b\) are greater than \(-2\) and less than \(5\). We don't include \(-2\) and \(5\) themselves in the solution set because neither inequality is "or equal to." To express this using interval notation, we use parentheses \(\) rather than brackets \[\] to show that endpoints are not included. Thus, the interval notation for \(-2 < b < 5\) is \((-2, 5)\).

If the endpoints were included, we'd use brackets, like this: \([-2, 5]\). But in this case, since \(-2\) and \(5\) are not part of the solution, the right representation is \((-2, 5)\). This is a crucial distinction to understand in interval notation.

Number Line Representation

Number lines are a fantastic visual tool to represent inequalities and understand the solution set. To illustrate the solution of our compound inequality \(-2 < b < 5\) on a number line, we follow a simple method.

First, draw a horizontal line and mark points that correspond to \(-2\) and \(5\). These are your boundary points. Since the inequality does not include these exact values, place open circles over \(-2\) and \(5\). Open circles signify that these points are not part of the solution.

Next, shade the region on the line between \(-2\) and \(5\). The shaded segment indicates all possible values of \(b\) that satisfy the compound inequality. The open circles clearly show that \(-2\) and \(5\) themselves are not included in the set. This graphical depiction enhances understanding of where \(b\) lies on the number line.

Solving Inequalities Step-by-Step

Solving compound inequalities like \(-2 < -b + 3 < 5\) involves breaking them down into simpler parts. Let's walk through it.

• First, separate the compound inequality into its two simpler inequalities, which are \(-2 < -b + 3\) and \(-b + 3 < 5\). Solve them independently.


• For \(-2 < -b + 3\), subtract 3 from both sides to isolate the \(-b\) term, leading to \(-5 < -b\). To solve for \(b\), multiply or divide by \(-1\), which requires flipping the inequality sign. This gives \(b < 5\).


• Next, for \(-b + 3 < 5\), subtract 3 from both sides to get \(-b < 2\). Solve for \(b\) by multiplying or dividing by \(-1\) and again flip the inequality sign. This results in \(b > -2\).


• Combine these solutions to get the compound inequality \(-2 < b < 5\).

This step-by-step approach breaks down complex problems into simpler segments, making them easier to solve and understand.