Solve For $k$:$4k + 18 \leq -2$

Introduction

In mathematics, solving inequalities is a crucial concept that helps us understand the relationship between different variables. In this article, we will focus on solving the inequality $4k + 18 \leq -2$, where $k$ is the variable we need to solve for. We will use algebraic methods to isolate the variable and find the solution set.

Understanding the Inequality

The given inequality is $4k + 18 \leq -2$. To solve this inequality, we need to isolate the variable $k$ on one side of the inequality sign. The inequality sign $\leq$ indicates that the expression on the left-hand side is less than or equal to the expression on the right-hand side.

Step 1: Subtract 18 from Both Sides

To isolate the term with the variable $k$, we need to subtract 18 from both sides of the inequality. This will help us get rid of the constant term on the left-hand side.

4k + 18 \leq -2
4k \leq -2 - 18
4k \leq -20

Step 2: Divide Both Sides by 4

Now that we have isolated the term with the variable $k$, we need to divide both sides of the inequality by 4. This will help us solve for $k$.

4k \leq -20
k \leq \frac{-20}{4}
k \leq -5

Conclusion

In this article, we solved the inequality $4k + 18 \leq -2$ by using algebraic methods. We first subtracted 18 from both sides of the inequality to isolate the term with the variable $k$. Then, we divided both sides of the inequality by 4 to solve for $k$. The solution to the inequality is $k \leq -5$.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always check the direction of the inequality sign.
• Use inverse operations to isolate the variable.
• Be careful when dividing or multiplying both sides of the inequality by a negative number.
• Check the solution set by plugging in values into the original inequality.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

• Business: Inequality solving is used to determine the maximum or minimum values of a company's profits or losses.
• Economics: Inequality solving is used to analyze the relationship between different economic variables, such as supply and demand.
• Science: Inequality solving is used to model real-world phenomena, such as population growth or chemical reactions.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

• Incorrectly applying inverse operations: Make sure to use the correct inverse operation to isolate the variable.
• Forgetting to check the direction of the inequality sign: Always check the direction of the inequality sign before solving the inequality.
• Not checking the solution set: Make sure to check the solution set by plugging in values into the original inequality.

Conclusion

Introduction

In our previous article, we solved the inequality $4k + 18 \leq -2$ by using algebraic methods. We first subtracted 18 from both sides of the inequality to isolate the term with the variable $k$. Then, we divided both sides of the inequality by 4 to solve for $k$. The solution to the inequality is $k \leq -5$. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to check the direction of the inequality sign. This will help you determine the correct method to use in solving the inequality.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable in an inequality, you need to use inverse operations. For example, if the inequality is $4k + 18 \leq -2$, you can subtract 18 from both sides to get $4k \leq -20$. Then, you can divide both sides by 4 to get $k \leq -5$.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the expression to determine the solution set. For example, if the inequality is $x^2 + 4x + 4 \leq 0$, you can factor the expression as $(x + 2)^2 \leq 0$. Then, you can use the sign of the expression to determine that the solution set is $x = -2$.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values of the variable that satisfy the inequality. For example, if the inequality is $x \leq 5$, the solution set is all values of $x$ that are less than or equal to 5.

Q: How do I check the solution set of an inequality?

A: To check the solution set of an inequality, you need to plug in values into the original inequality. For example, if the inequality is $x \leq 5$, you can plug in $x = 4$ and $x = 6$ to check that the solution set is all values of $x$ that are less than or equal to 5.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always check the direction of the inequality sign.
• Use inverse operations to isolate the variable.
• Be careful when dividing or multiplying both sides of the inequality by a negative number.
• Check the solution set by plugging in values into the original inequality.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

• Business: Inequality solving is used to determine the maximum or minimum values of a company's profits or losses.
• Economics: Inequality solving is used to analyze the relationship between different economic variables, such as supply and demand.
• Science: Inequality solving is used to model real-world phenomena, such as population growth or chemical reactions.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

• Incorrectly applying inverse operations: Make sure to use the correct inverse operation to isolate the variable.
• Forgetting to check the direction of the inequality sign: Always check the direction of the inequality sign before solving the inequality.
• Not checking the solution set: Make sure to check the solution set by plugging in values into the original inequality.

Conclusion

In conclusion, solving inequalities is a crucial concept in mathematics that helps us understand the relationship between different variables. By using algebraic methods, we can solve inequalities and find the solution set. Remember to check the direction of the inequality sign, use inverse operations to isolate the variable, and check the solution set by plugging in values into the original inequality. With practice and patience, you can become proficient in solving inequalities and apply them to real-world problems.