Start by isolating the variable term

Oct 13, 2025 by miodeya.com 102 views

Introduction

In mathematics, solving inequalities is a crucial skill that requires a deep understanding of algebraic manipulations and logical reasoning. In this article, we will explore how four individuals, Amelia, Luis, Shauna, and Clarence, approached solving the inequality $7.2b + 6.5 > 4.8b - 8.1$ . We will examine their respective methods, highlighting the strengths and weaknesses of each approach.

The Inequality

The given inequality is $7.2b + 6.5 > 4.8b - 8.1$ . This is a linear inequality, where the variable $b$ is the subject of the inequality.

Step 1: Subtracting $7.2b$ from Both Sides

Amelia started by subtracting $7.2b$ from both sides of the inequality to get $6.5 > -2.4b - 8.1$ . This step is a common first step in solving linear inequalities, as it isolates the variable term on one side of the inequality.

# Amelia's approach
inequality = "7.2b + 6.5 > 4.8b - 8.1"
amelia_inequality = "6.5 > -2.4b - 8.1"
print(amelia_inequality)

Step 2: Adding $8.1$ to Both Sides

Next, Amelia added $8.1$ to both sides of the inequality to get $14.6 > -2.4b$ . This step is necessary to isolate the variable term further.

# Amelia's approach
amelia_inequality = "6.5 > -2.4b - 8.1"
amelia_inequality = "14.6 > -2.4b"
print(amelia_inequality)

Step 3: Dividing Both Sides by $-2.4$

Finally, Amelia divided both sides of the inequality by $-2.4$ to get $b < -6.0833$ . This step is necessary to solve for the variable $b$ .

# Amelia's approach
amelia_inequality = "14.6 > -2.4b"
amelia_solution = "b < -6.0833"
print(amelia_solution)

Luis's Approach

Luis, on the other hand, started by subtracting $4.8b$ from both sides of the inequality to get $7.2b - 4.8b + 6.5 > -8.1$ . This step is a common first step in solving linear inequalities, as it isolates the variable term on one side of the inequality.

# Luis's approach
inequality = "7.2b + 6.5 > 4.8b - 8.1"
luis_inequality = "2.4b + 6.5 > -8.1"
print(luis_inequality)

Step 2: Subtracting $6.5$ from Both Sides

Next, Luis subtracted $6.5$ from both sides of the inequality to get $2.4b > -14.6$ . This step is necessary to isolate the variable term further.

# Luis's approach
luis_inequality = "2.4b + 6.5 > -8.1"
luis_inequality = "2.4b > -14.6"
print(luis_inequality)

Step 3: Dividing Both Sides by $2.4$

Finally, Luis divided both sides of the inequality by $2.4$ to get $b > -6.0833$ . This step is necessary to solve for the variable $b$ .

# Luis's approach
luis_inequality = "2.4b > -14.6"
luis_solution = "b > -6.0833"
print(luis_solution)

Shauna's Approach

Shauna, on the other hand, started by adding $8.1$ to both sides of the inequality to get $7.2b + 6.5 + 8.1 > 4.8b$ . This step is a common first step in solving linear inequalities, as it isolates the variable term on one side of the inequality.

# Shauna's approach
inequality = "7.2b + 6.5 > 4.8b - 8.1"
shauna_inequality = "7.2b + 6.5 + 8.1 > 4.8b"
print(shauna_inequality)

Step 2: Subtracting $4.8b$ from Both Sides

Next, Shauna subtracted $4.8b$ from both sides of the inequality to get $7.2b - 4.8b + 6.5 + 8.1 > 0$ . This step is necessary to isolate the variable term further.

# Shauna's approach
shauna_inequality = "7.2b + 6.5 + 8.1 > 4.8b"
shauna_inequality = "2.4b + 14.6 > 0"
print(shauna_inequality)

Step 3: Subtracting $14.6$ from Both Sides

Finally, Shauna subtracted $14.6$ from both sides of the inequality to get $2.4b > -14.6$ . This step is necessary to isolate the variable term further.

# Shauna's approach
shauna_inequality = "2.4b + 14.6 > 0"
shauna_inequality = "2.4b > -14.6"
print(shauna_inequality)

Step 4: Dividing Both Sides by $2.4$

Finally, Shauna divided both sides of the inequality by $2.4$ to get $b > -6.0833$ . This step is necessary to solve for the variable $b$ .

# Shauna's approach
shauna_inequality = "2.4b > -14.6"
shauna_solution = "b > -6.0833"
print(shauna_solution)

Clarence's Approach

Clarence, on the other hand, started by adding $8.1$ to both sides of the inequality to get $7.2b + 6.5 + 8.1 > 4.8b$ . This step is a common first step in solving linear inequalities, as it isolates the variable term on one side of the inequality.

# Clarence's approach
inequality = "7.2b + 6.5 > 4.8b - 8.1"
clarence_inequality = "7.2b + 6.5 + 8.1 > 4.8b"
print(clarence_inequality)

Step 2: Subtracting $4.8b$ from Both Sides

Next, Clarence subtracted $4.8b$ from both sides of the inequality to get $7.2b - 4.8b + 6.5 + 8.1 > 0$ . This step is necessary to isolate the variable term further.

# Clarence's approach
clarence_inequality = "7.2b + 6.5 + 8.1 > 4.8b"
clarence_inequality = "2.4b + 14.6 > 0"
print(clarence_inequality)

Step 3: Subtracting $14.6$ from Both Sides

Finally, Clarence subtracted $14.6$ from both sides of the inequality to get $2.4b > -14.6$ . This step is necessary to isolate the variable term further.

# Clarence's approach
clarence_inequality = "2.4b + 14.6 > 0"
clarence_inequality = "2.4b > -14.6"
print(clarence_inequality)

Step 4: Dividing Both Sides by $2.4$

Finally, Clarence divided both sides of the inequality by $2.4$ to get $b > -6.0833$ . This step is necessary to solve for the variable $b$ .

# Clarence's approach
clarence_inequality = "2.4b > -14.6"
clarence_solution = "b > -6.0833"
print(clarence_solution)

Conclusion

In conclusion, all four individuals, Amelia, Luis, Shauna, and Clarence, arrived at the same solution, $b > -6.0833$ . However, their approaches differed in the order of operations and the steps taken to isolate the variable term. This highlights the importance of understanding the underlying principles of algebraic manipulations and logical reasoning in solving inequalities.

Recommendations

Based on the analysis, we recommend the following:

Start by isolating the variable term: In all four approaches, the first step was to isolate the variable term on one side of the inequality. This is a crucial step in solving linear inequalities.
Q&A: Solving Inequalities - A Comparative Analysis =====================================================

Introduction

In our previous article, we explored how four individuals, Amelia, Luis, Shauna, and Clarence, approached solving the inequality $7.2b + 6.5 > 4.8b - 8.1$ . We examined their respective methods, highlighting the strengths and weaknesses of each approach. In this article, we will answer some frequently asked questions (FAQs) related to solving inequalities.

Q: What is the first step in solving a linear inequality?

A: The first step in solving a linear inequality is to isolate the variable term on one side of the inequality. This can be done by adding or subtracting the same value to both sides of the inequality.

Q: How do I isolate the variable term in a linear inequality?

A: To isolate the variable term, you can add or subtract the same value to both sides of the inequality. For example, if you have the inequality $7.2b + 6.5 > 4.8b - 8.1$ , you can add $8.1$ to both sides to get $7.2b + 6.5 + 8.1 > 4.8b$ .

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, whereas solving an inequality involves finding the range of values of the variable that make the inequality true.

Q: How do I determine the direction of the inequality?

A: The direction of the inequality is determined by the sign of the coefficient of the variable term. If the coefficient is positive, the inequality is greater than or equal to (≥), and if the coefficient is negative, the inequality is less than or equal to (≤).

Q: Can I use the same steps to solve a quadratic inequality as I would to solve a linear inequality?

A: No, you cannot use the same steps to solve a quadratic inequality as you would to solve a linear inequality. Quadratic inequalities require a different set of steps and techniques to solve.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you can use a number line or a coordinate plane. Plot a point on the number line or coordinate plane that satisfies the inequality, and then shade the region that satisfies the inequality.

Q: Can I use a calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, it's always a good idea to check your work by hand to ensure that the solution is correct.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include:

Not isolating the variable term on one side of the inequality
Not checking the direction of the inequality
Not considering the restrictions on the variable
Not checking the solution by plugging it back into the original inequality

Conclusion

In conclusion, solving linear inequalities requires a deep understanding of algebraic manipulations and logical reasoning. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear inequalities.

Additional Resources

For more information on solving linear inequalities, check out the following resources:

Khan Academy: Solving Linear Inequalities
Mathway: Solving Linear Inequalities
Wolfram Alpha: Solving Linear Inequalities

Practice Problems

Try solving the following linear inequalities:

$3x + 2 > 5$
$2x - 3 < 7$
$x + 4 > 2$
$x - 2 < 3$

Answer Key

$x > 1$
$x < 5$
$x > -2$
$x < 5$