Solve This Inequality 3b 7 32

How to Solve the Linear Inequality 3b + 7 < 32

Solving linear inequalities like 3b + 7 < 32 is a foundational skill in algebra that opens the door to understanding ranges of possible solutions rather than a single answer. Unlike equations that demand one precise value, inequalities describe a whole set of numbers that satisfy a condition—a concept essential for real-world problems involving budgets, measurements, and constraints. This guide will walk you through every step of solving 3b + 7 < 32, from the basic principles to verifying your answer and applying the knowledge practically. By the end, you will not only solve this specific problem but also possess a clear, repeatable method for tackling any similar linear inequality.

Understanding the Language of Inequalities

Before manipulating symbols, we must understand what an inequality expresses. The expression 3b + 7 < 32 uses the "less than" symbol (<). This statement is true for any value of b that makes the left side smaller than the right side. The solution is not a single number but an interval—a continuous range of values.

Key symbols to recognize:

< : less than
> : greater than
≤ : less than or equal to
≥ : greater than or equal to

Our goal is to isolate the variable b on one side of the inequality, following the same logical principles we use for equations. The core rule is: Whatever operation you perform on one side, you must perform on the other to maintain the balance or, in this case, the truth of the statement.

Step-by-Step Solution to 3b + 7 < 32

Let's solve the inequality systematically.

Step 1: Identify the operations applied to b. Looking at the left side, 3b + 7, we see that b is first multiplied by 3 and then 7 is added.

Step 2: Reverse the order of operations (PEMDAS/BODMAS) to isolate b. We undo the most recent operation first. The last thing done to b was adding 7. So, we do the opposite: subtract 7 from both sides.

3b + 7 - 7 < 32 - 7

This simplifies to:

3b < 25

Step 3: Undo the multiplication. Now, b is multiplied by 3. To undo this, we divide both sides by 3. Since 3 is a positive number, the inequality direction remains the same. (A critical rule: you only flip the inequality symbol <, >, ≤, ≥ when you multiply or divide by a negative number).

3b / 3 < 25 / 3

This gives us the solution:

b < 25/3 or b < 8.333...

Final Answer: b < 25/3 or, in decimal form, b < 8.333....

Visualizing the Solution: Number Line and Interval Notation

A solution like b < 25/3 is abstract. We make it concrete through visualization.

1. The Number Line: Draw a horizontal line. Mark the point 25/3 (approximately 8.33). Because the inequality is strictly "less than" (<) and not "less than or equal to" (≤), we use an open circle at 25/3 to show this point is not included in the solution. Then, shade or draw an arrow extending to the left from the open circle, indicating all numbers smaller than 25/3 are solutions.

2. Interval Notation: This is the standard mathematical shorthand. The solution b < 25/3 is written as:

(-∞, 25/3)

The parentheses ( ) mean the endpoint is not included. The -∞ symbol (negative infinity) indicates the solution extends indefinitely in the negative direction. There is no upper bound other than 25/3.

Verifying Your Solution: The Importance of Checking

Never trust an algebraic manipulation blindly. Always test values from your solution set and from outside it.

Test a value in the solution set: Choose a simple number less than 25/3, like b = 0.

3(0) + 7 < 32 → 0 + 7 < 32 → 7 < 32. This is TRUE. ✅
Test the boundary value: The boundary is b = 25/3.

3(25/3) + 7 < 32 → 25 + 7 < 32 → 32 < 32. This is FALSE, as expected, since < does not include equality. ✅
Test a value outside the solution set: Choose a number greater than 25/3, like b = 9.

3(9) + 7 < 32 → 27 + 7 < 32 → 34 < 32. This is FALSE. ✅

This check confirms our solution b < 25/3 is correct.

Common Mistakes and How to Avoid Them

Forgetting to Flip the Inequality Sign: This only happens when multiplying or dividing by a negative number. In our problem, we divided by +3, so no flip was needed. Example of error: If we had -3b + 7 < 32, subtracting 7 gives -3b < 25. Dividing