Consider The Following Compound Inequality:$-10 \leq X - 6 \ \textless \ -4$Solve The Inequality For $x$. - Answers Of The Form $3 \ \textless \ X$ And $x \ \textless \ 5$ Should Be Entered As $3 \

Introduction

In mathematics, inequalities are used to describe the relationship between two or more values. Compound inequalities, in particular, involve multiple inequalities that are combined using logical operators such as "and" or "or." In this article, we will focus on solving compound inequalities of the form $a \leq x - b < c$, where $a$, $b$, and $c$ are constants. We will use the given compound inequality $-10 \leq x - 6 < -4$ as an example to illustrate the steps involved in solving such inequalities.

Understanding the Compound Inequality

Before we proceed to solve the compound inequality, let's break it down and understand its components. The given inequality is $-10 \leq x - 6 < -4$. This can be read as "x minus 6 is greater than or equal to -10, and x minus 6 is less than -4." The inequality has two parts: the first part is $x - 6 \geq -10$, and the second part is $x - 6 < -4$.

Step 1: Add 6 to Both Sides of the Inequality

To solve the compound inequality, we need to isolate the variable $x$. We can start by adding 6 to both sides of the inequality. This will give us $x - 6 + 6 \geq -10 + 6$ and $x - 6 + 6 < -4 + 6$. Simplifying both sides, we get $x \geq -4$ and $x < 0$.

Step 2: Write the Solution in Interval Notation

The solution to the compound inequality is a combination of two inequalities: $x \geq -4$ and $x < 0$. We can write this solution in interval notation as $[-4, 0)$.

Step 3: Check the Solution

To ensure that our solution is correct, we need to check it by plugging in some values. Let's choose $x = -3$ and $x = 0$ to test our solution. If $x = -3$, then $x - 6 = -9$, which satisfies both inequalities. If $x = 0$, then $x - 6 = -6$, which also satisfies both inequalities. Therefore, our solution $[-4, 0)$ is correct.

Conclusion

Solving compound inequalities requires careful attention to detail and a step-by-step approach. By adding 6 to both sides of the inequality and writing the solution in interval notation, we can find the solution to the compound inequality $-10 \leq x - 6 < -4$. Remember to check your solution by plugging in some values to ensure that it is correct.

Example 1: Solving a Compound Inequality

Solve the compound inequality $-5 \leq x + 2 < 3$.

Step 1: Add -2 to Both Sides of the Inequality

To solve the compound inequality, we need to isolate the variable $x$. We can start by adding -2 to both sides of the inequality. This will give us $x + 2 - 2 \geq -5 - 2$ and $x + 2 - 2 < 3 - 2$. Simplifying both sides, we get $x \geq -7$ and $x < 1$.

Step 2: Write the Solution in Interval Notation

The solution to the compound inequality is a combination of two inequalities: $x \geq -7$ and $x < 1$. We can write this solution in interval notation as $[-7, 1)$.

Step 3: Check the Solution

To ensure that our solution is correct, we need to check it by plugging in some values. Let's choose $x = -6$ and $x = 0$ to test our solution. If $x = -6$, then $x + 2 = -4$, which satisfies both inequalities. If $x = 0$, then $x + 2 = 2$, which also satisfies both inequalities. Therefore, our solution $[-7, 1)$ is correct.

Example 2: Solving a Compound Inequality

Solve the compound inequality $-3 \leq x - 1 < 2$.

Step 1: Add 1 to Both Sides of the Inequality

To solve the compound inequality, we need to isolate the variable $x$. We can start by adding 1 to both sides of the inequality. This will give us $x - 1 + 1 \geq -3 + 1$ and $x - 1 + 1 < 2 + 1$. Simplifying both sides, we get $x \geq -2$ and $x < 3$.

Step 2: Write the Solution in Interval Notation

The solution to the compound inequality is a combination of two inequalities: $x \geq -2$ and $x < 3$. We can write this solution in interval notation as $[-2, 3)$.

Step 3: Check the Solution

To ensure that our solution is correct, we need to check it by plugging in some values. Let's choose $x = -1$ and $x = 0$ to test our solution. If $x = -1$, then $x - 1 = -2$, which satisfies both inequalities. If $x = 0$, then $x - 1 = -1$, which also satisfies both inequalities. Therefore, our solution $[-2, 3)$ is correct.

Tips and Tricks

• When solving compound inequalities, make sure to add or subtract the same value to both sides of the inequality.
• Use interval notation to write the solution to the compound inequality.
• Check your solution by plugging in some values to ensure that it is correct.

Conclusion

Q: What is a compound inequality?

A: A compound inequality is an inequality that involves multiple inequalities combined using logical operators such as "and" or "or." For example, the compound inequality $-10 \leq x - 6 < -4$ involves two inequalities: $x - 6 \geq -10$ and $x - 6 < -4$.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to isolate the variable by adding or subtracting the same value to both sides of the inequality. You can then write the solution in interval notation.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using a specific notation. For example, the solution to the compound inequality $-10 \leq x - 6 < -4$ can be written in interval notation as $[-4, 0)$.

Q: How do I check my solution to a compound inequality?

A: To check your solution to a compound inequality, you need to plug in some values to ensure that they satisfy both inequalities. For example, if you have the solution $[-4, 0)$, you can plug in $x = -3$ and $x = 0$ to test your solution.

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

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

• Adding or subtracting different values to both sides of the inequality
• Not writing the solution in interval notation
• Not checking the solution by plugging in some values

Q: Can I use a calculator to solve compound inequalities?

A: Yes, you can use a calculator to solve compound inequalities. However, it's always a good idea to check your solution by plugging in some values to ensure that it is correct.

Q: How do I graph a compound inequality on a number line?

A: To graph a compound inequality on a number line, you need to plot the solution in interval notation on the number line. For example, if the solution is $[-4, 0)$, you would plot a closed circle at $-4$ and an open circle at $0$.

Q: Can I use compound inequalities to solve systems of equations?

A: Yes, you can use compound inequalities to solve systems of equations. By combining the inequalities from each equation, you can find the solution to the system.

Q: What are some real-world applications of compound inequalities?

A: Compound inequalities have many real-world applications, including:

• Modeling population growth and decline
• Analyzing financial data
• Solving optimization problems

Conclusion

Solving compound inequalities requires careful attention to detail and a step-by-step approach. By understanding the basics of compound inequalities and practicing with examples, you can become proficient in solving these types of inequalities. Remember to check your solution by plugging in some values to ensure that it is correct. With practice and patience, you will become proficient in solving compound inequalities and be able to apply them to real-world problems.