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Compound Inequality: Solving for x

In mathematics, inequalities are used to describe relationships between variables. A compound inequality is a combination of two or more inequalities that are combined using logical operators such as "and" or "or". In this article, we will focus on solving a compound inequality of the form $-7 \leq x - 4 < 0$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Compound Inequality

The given compound inequality is $-7 \leq x - 4 < 0$. This means that the value of $x - 4$ is between $-7$ and $0$, including $-7$ but not including $0$. To solve this inequality, we need to isolate the variable $x$.

Step 1: Add 4 to Both Sides

The first step in solving the inequality is to add 4 to both sides of the inequality. This will help us isolate the variable $x$. When we add 4 to both sides, we get:

$$-7 + 4 \leq x - 4 + 4 < 0 + 4$$

Simplifying the left-hand side, we get:

$$-3 \leq x < 4$$

Step 2: Write the Solution in Interval Notation

The solution to the inequality is $-3 \leq x < 4$. We can write this in interval notation as $[-3, 4)$. This means that the value of $x$ can be any real number between $-3$ and $4$, including $-3$ but not including $4$.

Step 3: Check the Solution

To check the solution, we can plug in a value of $x$ that satisfies the inequality and see if it is true. Let's choose $x = 0$. Plugging this value into the original inequality, we get:

$$-7 \leq 0 - 4 < 0$$

Simplifying the left-hand side, we get:

$$-7 \leq -4 < 0$$

This is true, so the solution $x = 0$ satisfies the inequality.

In this article, we solved a compound inequality of the form $-7 \leq x - 4 < 0$. We broke down the solution step by step and provided a clear explanation of each step. We added 4 to both sides of the inequality to isolate the variable $x$, and then wrote the solution in interval notation as $[-3, 4)$. We also checked the solution by plugging in a value of $x$ that satisfies the inequality.

• When solving a compound inequality, it's essential to follow the order of operations and simplify the inequality step by step.
• When adding or subtracting a constant to both sides of an inequality, make sure to add or subtract the same constant to both sides.
• When writing the solution in interval notation, make sure to include the endpoints of the interval.

• When solving a compound inequality, it's easy to get confused and add or subtract the wrong constant to both sides. Make sure to double-check your work and simplify the inequality step by step.
• When writing the solution in interval notation, make sure to include the endpoints of the interval. Omitting the endpoints can lead to incorrect solutions.

Compound inequalities have many real-world applications in fields such as economics, finance, and engineering. For example, a company may want to determine the range of prices for a product that will result in a profit of at least $1000. In this case, the company can use a compound inequality to determine the range of prices that will satisfy the condition.

In conclusion, solving compound inequalities requires careful attention to detail and a step-by-step approach. By following the order of operations and simplifying the inequality step by step, we can arrive at the correct solution. We hope this article has provided a clear explanation of how to solve compound inequalities and has helped you to understand the concept better.
Compound Inequality: Q&A

In our previous article, we solved a compound inequality of the form $-7 \leq x - 4 < 0$. We broke down the solution step by step and provided a clear explanation of each step. In this article, we will answer some common questions that students often have when solving compound inequalities.

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities that are combined using logical operators such as "and" or "or". For example, $-7 \leq x - 4 < 0$ is a compound inequality that combines two inequalities: $-7 \leq x - 4$ and $x - 4 < 0$.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to follow the order of operations and simplify the inequality step by step. Here are the steps to solve a compound inequality:

1. Simplify the inequality by combining like terms.
2. Add or subtract a constant to both sides of the inequality.
3. Write the solution in interval notation.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality in a compact form. For example, the solution to the inequality $-3 \leq x < 4$ can be written in interval notation as $[-3, 4)$. This means that the value of $x$ can be any real number between $-3$ and $4$, including $-3$ but not including $4$.

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

A: To check the solution to a compound inequality, you need to plug in a value of $x$ that satisfies the inequality and see if it is true. For example, if the solution to the inequality $-7 \leq x - 4 < 0$ is $x = 0$, you can plug this value into the original inequality and see if it is true.

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

A: Here are some common mistakes to avoid when solving compound inequalities:

• Adding or subtracting the wrong constant to both sides of the inequality.
• Omitting the endpoints of the interval when writing the solution in interval notation.
• Not following the order of operations and simplifying the inequality step by step.

Q: How do I apply compound inequalities to real-world problems?

A: Compound inequalities have many real-world applications in fields such as economics, finance, and engineering. For example, a company may want to determine the range of prices for a product that will result in a profit of at least $1000. In this case, the company can use a compound inequality to determine the range of prices that will satisfy the condition.

Q: What are some tips for solving compound inequalities?

A: Here are some tips for solving compound inequalities:

• Follow the order of operations and simplify the inequality step by step.
• Add or subtract a constant to both sides of the inequality.
• Write the solution in interval notation.
• Check the solution by plugging in a value of $x$ that satisfies the inequality.

In conclusion, solving compound inequalities requires careful attention to detail and a step-by-step approach. By following the order of operations and simplifying the inequality step by step, we can arrive at the correct solution. We hope this article has provided a clear explanation of how to solve compound inequalities and has helped you to understand the concept better.