Which Is The Solution To The Inequality? 2 3 5 \textless B − 8 15 2 \frac{3}{5} \ \textless \ B - \frac{8}{15} 2 5 3 ​ \textless B − 15 8 ​ A. B \textless 2 1 15 B \ \textless \ 2 \frac{1}{15} B \textless 2 15 1 ​ B. B \textgreater 2 1 15 B \ \textgreater \ 2 \frac{1}{15} B \textgreater 2 15 1 ​ C. B \textless 3 2 15 B \ \textless \ 3 \frac{2}{15} B \textless 3 15 2 ​ D. $b \

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. Solving inequalities requires a different approach than solving equations, and it's essential to understand the rules and techniques involved. In this article, we will focus on solving the inequality $2 \frac{3}{5} \ \textless \ b - \frac{8}{15}$ and explore the different solution options.

Understanding the Inequality

The given inequality is $2 \frac{3}{5} \ \textless \ b - \frac{8}{15}$. To solve this inequality, we need to isolate the variable $b$ and determine the range of values that satisfy the inequality.

Step 1: Convert Mixed Numbers to Improper Fractions

The first step is to convert the mixed numbers to improper fractions. We can convert $2 \frac{3}{5}$ to an improper fraction by multiplying the whole number by the denominator and adding the numerator.

$2 \frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{13}{5}$

Similarly, we can convert $b - \frac{8}{15}$ to an improper fraction by subtracting $\frac{8}{15}$ from $b$.

$b - \frac{8}{15} = \frac{15b - 8}{15}$

Step 2: Rewrite the Inequality

Now that we have converted the mixed numbers to improper fractions, we can rewrite the inequality as:

$\frac{13}{5} \ \textless \ \frac{15b - 8}{15}$

Step 3: Multiply Both Sides by 15

To eliminate the fractions, we can multiply both sides of the inequality by 15.

$15 \times \frac{13}{5} \ \textless \ 15 \times \frac{15b - 8}{15}$

This simplifies to:

$39 \ \textless \ 15b - 8$

Step 4: Add 8 to Both Sides

To isolate the term with the variable, we can add 8 to both sides of the inequality.

$39 + 8 \ \textless \ 15b - 8 + 8$

This simplifies to:

$47 \ \textless \ 15b$

Step 5: Divide Both Sides by 15

Finally, we can divide both sides of the inequality by 15 to solve for $b$.

$\frac{47}{15} \ \textless \ \frac{15b}{15}$

This simplifies to:

$\frac{47}{15} \ \textless \ b$

Conclusion

The solution to the inequality $2 \frac{3}{5} \ \textless \ b - \frac{8}{15}$ is $b \ \textless \ \frac{47}{15}$. This means that the value of $b$ must be less than $\frac{47}{15}$ to satisfy the inequality.

Comparing the Solution Options

Now that we have solved the inequality, we can compare the solution options.

A. $b \ \textless \ 2 \frac{1}{15}$

B. $b \ \textgreater \ 2 \frac{1}{15}$

C. $b \ \textless \ 3 \frac{2}{15}$

D. $b \ \textgreater \ 3 \frac{2}{15}$

The correct solution option is C. $b \ \textless \ 3 \frac{2}{15}$, which is equivalent to $b \ \textless \ \frac{47}{15}$.

Final Answer

$$

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two or more expressions using a relation such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing operations that do not change the direction of the inequality. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

A: Solving an inequality is similar to solving an equation, but with a few key differences. When solving an inequality, you need to consider the direction of the inequality and ensure that the operations you perform do not change the direction.

Q: How do I know which direction to change the inequality when multiplying or dividing both sides?

A: When multiplying or dividing both sides of an inequality by a negative value, you need to change the direction of the inequality. For example, if you have the inequality $x > 5$ and you multiply both sides by -1, the inequality becomes $x < -5$.

Q: Can I add or subtract the same value from both sides of an inequality?

A: Yes, you can add or subtract the same value from both sides of an inequality without changing the direction of the inequality.

Q: Can I multiply or divide both sides of an inequality by the same value?

A: Yes, you can multiply or divide both sides of an inequality by the same value, but you need to be careful not to change the direction of the inequality if the value is negative.

Q: How do I solve an inequality with fractions?

A: To solve an inequality with fractions, you need to eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: Can I use the same steps to solve an inequality with decimals?

A: Yes, you can use the same steps to solve an inequality with decimals. However, you may need to round the decimal values to simplify the calculations.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you can plug in a value from the solution set into the original inequality and verify that it is true.

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

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

• Changing the direction of the inequality when multiplying or dividing both sides by a negative value
• Adding or subtracting the same value from both sides of an inequality without considering the direction of the inequality
• Multiplying or dividing both sides of an inequality by a value that is not the same on both sides
• Failing to check the solution to an inequality

Conclusion

Solving inequalities requires a different approach than solving equations, but with practice and patience, you can master the techniques and become proficient in solving inequalities. Remember to always check your solution to an inequality and avoid common mistakes to ensure that your answer is correct.