Solve For $k$.$-\frac{3}{5} \leq -3k$

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and solve problems. In this article, we will focus on solving inequalities involving fractions and variables. Specifically, we will solve the inequality $-\frac{3}{5} \leq -3k$ to find the value of $k$. We will break down the solution into manageable steps, using algebraic manipulations and logical reasoning.

Understanding the Inequality

The given inequality is $-\frac{3}{5} \leq -3k$. This means that the value of $-3k$ is greater than or equal to $-\frac{3}{5}$. To solve for $k$, we need to isolate the variable $k$ on one side of the inequality.

Step 1: Divide Both Sides by $-3$

To isolate $k$, we need to get rid of the coefficient $-3$ that is multiplied by $k$. We can do this by dividing both sides of the inequality by $-3$. However, when we divide or multiply both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

-\frac{3}{5} \leq -3k
\Rightarrow \frac{3}{5} \geq k

Step 2: Multiply Both Sides by $-1$

To make the inequality more manageable, we can multiply both sides by $-1$. This will change the direction of the inequality sign again.

\frac{3}{5} \geq k
\Rightarrow -\frac{3}{5} \leq k

Step 3: Multiply Both Sides by $5$

To eliminate the fraction, we can multiply both sides of the inequality by $5$. This will give us a more straightforward inequality.

-\frac{3}{5} \leq k
\Rightarrow -3 \leq 5k

Step 4: Divide Both Sides by $5$

Finally, we can divide both sides of the inequality by $5$ to solve for $k$.

-3 \leq 5k
\Rightarrow -\frac{3}{5} \leq k

Conclusion

In this article, we solved the inequality $-\frac{3}{5} \leq -3k$ to find the value of $k$. We broke down the solution into manageable steps, using algebraic manipulations and logical reasoning. By following these steps, we were able to isolate the variable $k$ and find its value. This problem illustrates the importance of understanding inequalities and how to solve them in mathematics.

Tips and Tricks

• When solving inequalities, it's essential to remember that when dividing or multiplying both sides by a negative number, we need to reverse the direction of the inequality sign.
• To eliminate fractions, we can multiply both sides of the inequality by the denominator.
• To solve for a variable, we need to isolate it on one side of the inequality.

Common Mistakes

• Failing to reverse the direction of the inequality sign when dividing or multiplying both sides by a negative number.
• Not eliminating fractions before solving for a variable.
• Not checking the solution to ensure it satisfies the original inequality.

Real-World Applications

Solving inequalities is a crucial skill in mathematics that has numerous real-world applications. For example:

• In finance, inequalities are used to calculate interest rates and investment returns.
• In science, inequalities are used to model population growth and chemical reactions.
• In engineering, inequalities are used to design and optimize systems.

Introduction

In our previous article, we solved the inequality $-\frac{3}{5} \leq -3k$ to find the value of $k$. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities. We will cover common questions, tips, and tricks to help you master this essential skill in mathematics.

Q: What is an inequality?

A: An inequality is a statement that compares two values or expressions using a mathematical symbol, such as $<$, $>$, $\leq$, or $\geq$. Inequalities are used to describe relationships between values or expressions that are not equal.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by a number or expression. However, when dividing or multiplying both sides by a negative number, you need to reverse the direction of the inequality sign.

Q: What is the difference between $<$ and $\leq$?

A: The symbol $<$ means "less than," while the symbol $\leq$ means "less than or equal to." For example, $x < 5$ means that $x$ is less than 5, while $x \leq 5$ means that $x$ is less than or equal to 5.

Q: How do I eliminate fractions in an inequality?

A: To eliminate fractions in an inequality, you can multiply both sides of the inequality by the denominator. For example, if you have the inequality $\frac{x}{2} \leq 3$, you can multiply both sides by 2 to get $x \leq 6$.

Q: What is the order of operations for solving inequalities?

A: The order of operations for solving inequalities is the same as for solving equations:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: Yes, you can add or subtract the same value to both sides of an inequality. For example, if you have the inequality $x < 5$, you can add 2 to both sides to get $x + 2 < 7$.

Q: Can I multiply or divide both sides of an inequality by a negative number?

A: Yes, you can multiply or divide both sides of an inequality by a negative number. However, when you do this, you need to reverse the direction of the inequality sign. For example, if you have the inequality $x > 5$, you can multiply both sides by $-1$ to get $-x < -5$.

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

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

• Failing to reverse the direction of the inequality sign when dividing or multiplying both sides by a negative number.
• Not eliminating fractions before solving for a variable.
• Not checking the solution to ensure it satisfies the original inequality.

Conclusion

Solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By mastering the art of solving inequalities, you will be better equipped to tackle complex problems and make informed decisions in various fields. We hope this Q&A guide has helped you better understand the concept of solving inequalities and has provided you with the tools and confidence to tackle more challenging problems.