What Is The Solution Set Of This Compound Inequality?${1 \leq |x+3| \leq 4}$Use The Drawing Tool(s) To Form The Correct Answer On The Provided Number Line.

Introduction

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." In this case, we have a compound inequality that involves absolute value. The given compound inequality is $1 \leq |x+3| \leq 4$. Our goal is to find the solution set of this compound inequality, which represents all the values of $x$ that satisfy the given inequality.

Understanding Absolute Value

Before we proceed, let's recall the definition of absolute value. The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line. In other words, it is the magnitude of $a$ without considering its direction. For example, $|5| = 5$ and $|-5| = 5$. This means that the absolute value of a number is always non-negative.

Breaking Down the Compound Inequality

The given compound inequality is $1 \leq |x+3| \leq 4$. To solve this inequality, we need to break it down into two separate inequalities:

1. $1 \leq |x+3|$
2. $|x+3| \leq 4$

Solving the First Inequality

Let's start by solving the first inequality: $1 \leq |x+3|$. To solve this inequality, we need to consider two cases:

Case 1: $x+3 \geq 0$

If $x+3 \geq 0$, then $|x+3| = x+3$. In this case, the inequality becomes $1 \leq x+3$, which simplifies to $x \geq -2$.

Case 2: $x+3 < 0$

If $x+3 < 0$, then $|x+3| = -(x+3)$. In this case, the inequality becomes $1 \leq -(x+3)$, which simplifies to $x \leq -4$.

Solving the Second Inequality

Now, let's solve the second inequality: $|x+3| \leq 4$. Again, we need to consider two cases:

Case 1: $x+3 \geq 0$

If $x+3 \geq 0$, then $|x+3| = x+3$. In this case, the inequality becomes $x+3 \leq 4$, which simplifies to $x \leq 1$.

Case 2: $x+3 < 0$

If $x+3 < 0$, then $|x+3| = -(x+3)$. In this case, the inequality becomes $-(x+3) \leq 4$, which simplifies to $x \geq -7$.

Combining the Results

Now that we have solved both inequalities, we need to combine the results. From the first inequality, we have $x \geq -2$ or $x \leq -4$. From the second inequality, we have $x \leq 1$ or $x \geq -7$. To find the solution set, we need to combine these two sets of inequalities.

Solution Set

The solution set of the compound inequality $1 \leq |x+3| \leq 4$ is the set of all values of $x$ that satisfy both inequalities. Combining the results, we get:

$x \geq -7$ and $x \leq 1$

This means that the solution set is the interval $[-7, 1]$.

Conclusion

In this article, we have solved the compound inequality $1 \leq |x+3| \leq 4$. We broke down the inequality into two separate inequalities and solved each one separately. We then combined the results to find the solution set, which is the interval $[-7, 1]$.

Drawing the Solution Set on a Number Line

To visualize the solution set, we can draw it on a number line. The solution set is the interval $[-7, 1]$, which includes all the values of $x$ between $-7$ and $1$, inclusive.

Here is the number line with the solution set marked:

• -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1

The solution set is the shaded region, which includes all the values of $x$ between $-7$ and $1$, inclusive.

Final Answer

The final answer is $\boxed{[-7, 1]}$.

Introduction

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." In this article, we have solved the compound inequality $1 \leq |x+3| \leq 4$ and found the solution set to be the interval $[-7, 1]$. However, we know that there are many more questions that students and teachers may have about compound inequalities. In this article, we will answer some of the most frequently asked questions (FAQs) about compound inequalities.

Q&A

Q1: What is a compound inequality?

A1: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or." For example, $1 \leq |x+3| \leq 4$ is a compound inequality.

Q2: How do I solve a compound inequality?

A2: To solve a compound inequality, you need to break it down into two separate inequalities and solve each one separately. Then, you need to combine the results to find the solution set.

Q3: What is the solution set of a compound inequality?

A3: The solution set of a compound inequality is the set of all values of the variable that satisfy the inequality. It is usually represented as an interval on a number line.

Q4: How do I graph a compound inequality on a number line?

A4: To graph a compound inequality on a number line, you need to draw a number line and mark the solution set with a shaded region. The solution set is the interval that includes all the values of the variable that satisfy the inequality.

Q5: What is the difference between a compound inequality and a single inequality?

A5: A single inequality is a single statement that compares two expressions, such as $x > 2$. A compound inequality, on the other hand, is a combination of two or more inequalities joined by the words "and" or "or."

Q6: Can I use the same method to solve a compound inequality with absolute value as I would with a single inequality?

A6: No, you cannot use the same method to solve a compound inequality with absolute value as you would with a single inequality. You need to break down the compound inequality into two separate inequalities and solve each one separately.

Q7: How do I know which method to use to solve a compound inequality?

A7: To determine which method to use to solve a compound inequality, you need to look at the inequality and determine whether it involves absolute value or not. If it involves absolute value, you need to break it down into two separate inequalities and solve each one separately.

Q8: Can I use a calculator to solve a compound inequality?

A8: Yes, you can use a calculator to solve a compound inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function to solve the inequality.

Q9: How do I check my answer to a compound inequality?

A9: To check your answer to a compound inequality, you need to plug in values from the solution set into the original inequality and make sure that they satisfy the inequality.

Q10: Can I use a graphing calculator to graph a compound inequality?

A10: Yes, you can use a graphing calculator to graph a compound inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function to graph the inequality.

Conclusion

In this article, we have answered some of the most frequently asked questions (FAQs) about compound inequalities. We have covered topics such as what a compound inequality is, how to solve a compound inequality, and how to graph a compound inequality on a number line. We hope that this article has been helpful in answering your questions about compound inequalities.

Final Answer

The final answer is $\boxed{[-7, 1]}$.