Which Graph Represents The Solution Set Of The Compound Inequality − 4 ≤ 3 X − 1 -4 \leq 3x - 1 − 4 ≤ 3 X − 1 And 2 X + 4 ≤ 18 2x + 4 \leq 18 2 X + 4 ≤ 18 ?

Introduction

Compound inequalities are a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will focus on solving compound inequalities of the form $a \leq f(x) \leq b$ and $c \leq g(x) \leq d$, where $f(x)$ and $g(x)$ are linear functions. We will use the compound inequality $-4 \leq 3x - 1$ and $2x + 4 \leq 18$ as an example to illustrate the steps involved in solving compound inequalities.

Understanding the Compound Inequality

A compound inequality is a statement that combines two or more inequalities using logical operators. In the given compound inequality, we have two inequalities:

1. $-4 \leq 3x - 1$
2. $2x + 4 \leq 18$

The logical operator "and" is used to connect these two inequalities, indicating that both inequalities must be true simultaneously.

Solving the First Inequality

To solve the first inequality, we need to isolate the variable $x$. We can do this by adding $1$ to both sides of the inequality:

$$-4 + 1 \leq 3x - 1 + 1$$

This simplifies to:

$$-3 \leq 3x$$

Next, we can divide both sides of the inequality by $3$ to solve for $x$:

$$\frac{-3}{3} \leq \frac{3x}{3}$$

This simplifies to:

$$-1 \leq x$$

Solving the Second Inequality

To solve the second inequality, we need to isolate the variable $x$. We can do this by subtracting $4$ from both sides of the inequality:

$$2x + 4 - 4 \leq 18 - 4$$

This simplifies to:

$$2x \leq 14$$

Next, we can divide both sides of the inequality by $2$ to solve for $x$:

$$\frac{2x}{2} \leq \frac{14}{2}$$

This simplifies to:

$$x \leq 7$$

Combining the Solutions

Now that we have solved both inequalities, we need to combine the solutions to find the solution set of the compound inequality. The solution set is the set of all values of $x$ that satisfy both inequalities.

From the first inequality, we have:

$$-1 \leq x$$

From the second inequality, we have:

$$x \leq 7$$

To find the solution set, we need to find the intersection of these two sets. The intersection of two sets is the set of all elements that are common to both sets.

In this case, the solution set is the set of all values of $x$ that satisfy both inequalities:

$$-1 \leq x \leq 7$$

Graphing the Solution Set

To graph the solution set, we need to plot the two inequalities on a number line. The first inequality is $-1 \leq x$, which can be represented by a closed circle at $x = -1$ and an open circle at $x = \infty$. The second inequality is $x \leq 7$, which can be represented by a closed circle at $x = 7$ and an open circle at $x = -\infty$.

The solution set is the intersection of these two sets, which is the set of all values of $x$ that satisfy both inequalities. This can be represented by a closed circle at $x = -1$ and an open circle at $x = 7$.

Conclusion

In this article, we have solved the compound inequality $-4 \leq 3x - 1$ and $2x + 4 \leq 18$ using a step-by-step approach. We have isolated the variable $x$ in each inequality, combined the solutions to find the solution set, and graphed the solution set on a number line. The solution set is the set of all values of $x$ that satisfy both inequalities, which is $-1 \leq x \leq 7$.

Example Problems

1. Solve the compound inequality $2x + 5 \leq 11$ and $x - 3 \leq 2$.
2. Solve the compound inequality $x + 2 \leq 9$ and $3x - 2 \leq 14$.
3. Solve the compound inequality $4x - 2 \leq 10$ and $2x + 1 \leq 13$.

Answer Key

1. $-3 \leq x \leq 3$
2. $3 \leq x \leq 6$
3. $2 \leq x \leq 6$

Tips and Tricks

• When solving compound inequalities, make sure to isolate the variable $x$ in each inequality.
• Combine the solutions to find the solution set.
• Graph the solution set on a number line to visualize the solution set.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Intermediate Algebra" by Charles P. McKeague
• [3] "College Algebra" by James Stewart
  Compound Inequality Q&A ==========================

Q: What is a compound inequality?

A: A compound inequality is a statement that combines two or more inequalities using logical operators. In the given compound inequality, we have two inequalities:

1. $-4 \leq 3x - 1$
2. $2x + 4 \leq 18$

The logical operator "and" is used to connect these two inequalities, indicating that both inequalities must be true simultaneously.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to follow these steps:

1. Solve each inequality separately.
2. Combine the solutions to find the solution set.
3. Graph the solution set on a number line.

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

A: A single inequality is a statement that compares two expressions using a single inequality symbol (e.g., $\leq$, $\geq$, $<$, or $>$). A compound inequality, on the other hand, combines two or more inequalities using logical operators (e.g., "and" or "or").

Q: Can I use the same method to solve a compound inequality with "or" as I do with "and"?

A: No, you cannot use the same method to solve a compound inequality with "or" as you do with "and". When using "or", you need to find the union of the solution sets, whereas when using "and", you need to find the intersection of the solution sets.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to plot the two inequalities on a number line. The first inequality is represented by a closed circle at the lower bound and an open circle at the upper bound. The second inequality is represented by a closed circle at the upper bound and an open circle at the lower bound. The solution set is the intersection of these two sets.

Q: Can I use a graphing calculator to solve a compound inequality?

A: Yes, you can use a graphing calculator to solve a compound inequality. Simply graph the two inequalities on the calculator and find the intersection of the two graphs.

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

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

• Not isolating the variable $x$ in each inequality.
• Not combining the solutions to find the solution set.
• Not graphing the solution set on a number line.

Q: Can I use compound inequalities to solve real-world problems?

A: Yes, you can use compound inequalities to solve real-world problems. Compound inequalities can be used to model situations where there are multiple constraints or conditions that must be met.

Q: What are some examples of real-world problems that can be solved using compound inequalities?

A: Some examples of real-world problems that can be solved using compound inequalities include:

• Finding the range of values for a variable that satisfies multiple conditions.
• Modeling the behavior of a system with multiple constraints.
• Solving optimization problems with multiple constraints.

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

A: Yes, you can use compound inequalities to solve systems of linear equations. By converting the system of linear equations into a compound inequality, you can use the methods for solving compound inequalities to find the solution set.

Q: What are some tips for solving compound inequalities?

A: Some tips for solving compound inequalities include:

• Read the problem carefully and understand what is being asked.
• Break down the problem into smaller, more manageable parts.
• Use a step-by-step approach to solve the compound inequality.
• Check your work to ensure that the solution set is correct.