Solve And Write Interval Notation For The Solution Set. Then Graph The Solution Set.$2x + 1 \leq -6 \quad \text{or} \quad 2x + 1 \geq 6$Select The Correct Choice Below And Fill In Any Answer Boxes In Your Choice.A. The Solution Set Is

Introduction

In this article, we will explore the process of solving and writing interval notation for the solution set of a given inequality. We will also learn how to graph the solution set. The inequality we will be working with is $2x + 1 \leq -6 \quad \text{or} \quad 2x + 1 \geq 6$. Our goal is to find the solution set and represent it in interval notation.

Step 1: Solve the Inequality

To solve the inequality, we need to isolate the variable $x$. We can start by subtracting $1$ from both sides of the inequality:

$$2x + 1 - 1 \leq -6 - 1$$

This simplifies to:

$$2x \leq -7$$

Next, we can divide both sides of the inequality by $2$:

$$\frac{2x}{2} \leq \frac{-7}{2}$$

This simplifies to:

$$x \leq -\frac{7}{2}$$

Now, let's consider the second part of the inequality: $2x + 1 \geq 6$. We can solve this inequality in a similar way:

$$2x + 1 - 1 \geq 6 - 1$$

This simplifies to:

$$2x \geq 5$$

Next, we can divide both sides of the inequality by $2$:

$$\frac{2x}{2} \geq \frac{5}{2}$$

This simplifies to:

$$x \geq \frac{5}{2}$$

Step 2: Write the Solution Set in Interval Notation

Now that we have solved the inequality, we can write the solution set in interval notation. The solution set is the set of all values of $x$ that satisfy the inequality. In this case, the solution set is the union of two intervals: $x \leq -\frac{7}{2}$ and $x \geq \frac{5}{2}$.

We can write the solution set in interval notation as:

$$\left(-\infty, -\frac{7}{2}\right] \cup \left[\frac{5}{2}, \infty\right)$$

Step 3: Graph the Solution Set

To graph the solution set, we need to plot the two intervals on a number line. The first interval, $x \leq -\frac{7}{2}$, includes all values of $x$ that are less than or equal to $-\frac{7}{2}$. The second interval, $x \geq \frac{5}{2}$, includes all values of $x$ that are greater than or equal to $\frac{5}{2}$.

We can graph the solution set by plotting the two intervals on a number line. The resulting graph will be a union of two intervals, with the first interval extending to the left and the second interval extending to the right.

Conclusion

In this article, we learned how to solve and write interval notation for the solution set of a given inequality. We also learned how to graph the solution set. The inequality we worked with was $2x + 1 \leq -6 \quad \text{or} \quad 2x + 1 \geq 6$. Our goal was to find the solution set and represent it in interval notation.

We found that the solution set is the union of two intervals: $x \leq -\frac{7}{2}$ and $x \geq \frac{5}{2}$. We wrote the solution set in interval notation as:

$$\left(-\infty, -\frac{7}{2}\right] \cup \left[\frac{5}{2}, \infty\right)$$

We also graphed the solution set by plotting the two intervals on a number line. The resulting graph is a union of two intervals, with the first interval extending to the left and the second interval extending to the right.

Final Answer

The final answer is:

\left(-\infty, -\frac{7}{2}\right] \cup \left[\frac{5}{2}, \infty\right)$<br/> **Solving and Graphing Inequalities in Interval Notation: Q&A** =========================================================== **Introduction** --------------- In our previous article, we explored the process of solving and writing interval notation for the solution set of a given inequality. We also learned how to graph the solution set. In this article, we will answer some common questions related to solving and graphing inequalities in interval notation. **Q: What is interval notation?** ----------------------------- A: Interval notation is a way of representing a set of numbers using a specific notation. It is commonly used to represent the solution set of an inequality. In interval notation, a set of numbers is represented as a range of values, with the lower and upper bounds of the range indicated by square brackets or parentheses. **Q: How do I determine the direction of the inequality symbol?** --------------------------------------------------------- A: When solving an inequality, the direction of the inequality symbol depends on the operation being performed. If the operation is addition or subtraction, the direction of the inequality symbol remains the same. However, if the operation is multiplication or division, the direction of the inequality symbol is reversed. **Q: What is the difference between a closed interval and an open interval?** ------------------------------------------------------------------- A: A closed interval is an interval that includes the endpoints, while an open interval is an interval that does not include the endpoints. In interval notation, a closed interval is represented by square brackets, while an open interval is represented by parentheses. **Q: How do I graph an inequality in interval notation?** --------------------------------------------------- A: To graph an inequality in interval notation, you need to plot the interval on a number line. If the interval is closed, you need to include the endpoint. If the interval is open, you need to exclude the endpoint. **Q: What is the union of two intervals?** ----------------------------------------- A: The union of two intervals is a set of numbers that includes all the numbers in both intervals. In interval notation, the union of two intervals is represented by the union symbol, which is a "U" with a horizontal line above it. **Q: How do I find the intersection of two intervals?** --------------------------------------------------- A: The intersection of two intervals is a set of numbers that includes all the numbers that are common to both intervals. In interval notation, the intersection of two intervals is represented by the intersection symbol, which is an "∩" symbol. **Q: What is the difference between a linear inequality and a nonlinear inequality?** ------------------------------------------------------------------------- A: A linear inequality is an inequality that can be written in the form ax + b ≤ c, where a, b, and c are constants. A nonlinear inequality is an inequality that cannot be written in this form. **Q: How do I solve a system of linear inequalities?** --------------------------------------------------- A: To solve a system of linear inequalities, you need to find the solution set of each inequality and then find the intersection of the solution sets. **Q: What is the importance of interval notation in mathematics?** --------------------------------------------------------- A: Interval notation is an important concept in mathematics because it provides a way to represent the solution set of an inequality in a concise and precise manner. It is used in a variety of mathematical applications, including algebra, calculus, and statistics. **Conclusion** ---------- In this article, we answered some common questions related to solving and graphing inequalities in interval notation. We hope that this article has provided you with a better understanding of the concept of interval notation and how it is used in mathematics. **Final Answer** -------------- The final answer is: Interval notation is a way of representing a set of numbers using a specific notation, and it is commonly used to represent the solution set of an inequality.