Solve And Write Interval Notation For The Solution Set. Then Graph The Solution Set.$-10 \ \textless \ 2x + 5 \ \textless \ 7$Select The Correct Choice Below And Fill In Any Answer Boxes In Your Choice:A. The Solution Set Is

Introduction

In mathematics, inequalities are used to describe relationships between variables. Solving and graphing inequalities in interval notation is an essential skill in algebra and mathematics. In this article, we will learn how to solve and write interval notation for the solution set, and then graph the solution set.

Step 1: Solve the Inequality

The given inequality is $-10 < 2x + 5 < 7$. To solve this inequality, we need to isolate the variable $x$.

Subtract 5 from all three parts of the inequality

Subtracting 5 from all three parts of the inequality gives us:

$$-10 - 5 < 2x + 5 - 5 < 7 - 5$$

This simplifies to:

$$-15 < 2x < 2$$

Divide all three parts of the inequality by 2

Dividing all three parts of the inequality by 2 gives us:

$$\frac{-15}{2} < \frac{2x}{2} < \frac{2}{2}$$

This simplifies to:

$$-7.5 < x < 1$$

Step 2: Write Interval Notation

Interval notation is a way of writing the solution set of an inequality. The solution set is the set of all values of $x$ that satisfy the inequality.

Write the solution set in interval notation

The solution set is the set of all values of $x$ that satisfy the inequality $-7.5 < x < 1$. This can be written in interval notation as:

$$(-7.5, 1)$$

Step 3: Graph the Solution Set

Graphing the solution set is an important step in understanding the inequality. The solution set is the set of all values of $x$ that satisfy the inequality.

Graph the solution set on a number line

To graph the solution set, we need to plot the points $-7.5$ and $1$ on a number line. The solution set is the set of all values of $x$ that lie between these two points.

The graph of the solution set is a line segment that extends from $-7.5$ to $1$.

Conclusion

In this article, we learned how to solve and write interval notation for the solution set, and then graph the solution set. We started by solving the inequality $-10 < 2x + 5 < 7$, and then wrote the solution set in interval notation as $(-7.5, 1)$. Finally, we graphed the solution set on a number line.

Answer

The solution set is $(-7.5, 1)$.

Discussion

This problem requires the student to solve an inequality and write the solution set in interval notation. The student must also graph the solution set on a number line. This problem is an important step in understanding inequalities and interval notation.

Key Concepts

• Solving inequalities
• Writing interval notation
• Graphing the solution set on a number line

Mathematical Operations

• Subtracting 5 from all three parts of the inequality
• Dividing all three parts of the inequality by 2

Mathematical Properties

• The properties of inequalities, such as the addition and subtraction properties
• The properties of interval notation, such as the union and intersection properties
  Solving and Graphing Inequalities in Interval Notation: Q&A ===========================================================

Introduction

In our previous article, we learned how to solve and write interval notation for the solution set, and then 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 writing the solution set of an inequality. It is a shorthand way of writing the set of all values of $x$ that satisfy the inequality.

Q: How do I write interval notation for a solution set?

A: To write interval notation for a solution set, you need to follow these steps:

1. Identify the solution set.
2. Determine the endpoints of the solution set.
3. Use the correct notation to indicate whether the endpoints are included or excluded.

For example, if the solution set is $(-7.5, 1)$, the interval notation would be $(-7.5, 1)$.

Q: What is the difference between open and closed intervals?

A: Open intervals are denoted by parentheses, and they indicate that the endpoints are excluded. Closed intervals are denoted by square brackets, and they indicate that the endpoints are included.

For example, the interval $(-7.5, 1)$ is an open interval, while the interval $[-7.5, 1]$ is a closed interval.

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

A: To graph an inequality on a number line, you need to follow these steps:

1. Identify the solution set.
2. Plot the endpoints of the solution set on the number line.
3. Use a line segment to indicate the solution set.

For example, if the solution set is $(-7.5, 1)$, you would plot the points $-7.5$ and $1$ on the number line, and then draw a line segment to indicate the solution set.

Q: What is the union and intersection of intervals?

A: The union of two intervals is the set of all values that are in either of the two intervals. The intersection of two intervals is the set of all values that are in both of the two intervals.

For example, if the two intervals are $(-7.5, 1)$ and $(2, 5)$, the union would be $(-7.5, 5)$, and the intersection would be $\emptyset$.

Q: How do I solve a compound inequality?

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

1. Identify the compound inequality.
2. Use the properties of inequalities to simplify the compound inequality.
3. Write the solution set in interval notation.

For example, if the compound inequality is $-10 < 2x + 5 < 7$, you would first simplify the inequality to $-7.5 < x < 1$, and then write the solution set in interval notation as $(-7.5, 1)$.

Conclusion

In this article, we answered some common questions related to solving and graphing inequalities in interval notation. We covered topics such as interval notation, open and closed intervals, graphing inequalities on a number line, and solving compound inequalities.

Key Concepts

• Interval notation
• Open and closed intervals
• Graphing inequalities on a number line
• Union and intersection of intervals
• Solving compound inequalities

Mathematical Operations

• Subtracting 5 from all three parts of the inequality
• Dividing all three parts of the inequality by 2
• Using the properties of inequalities to simplify the compound inequality

Mathematical Properties

• The properties of inequalities, such as the addition and subtraction properties
• The properties of interval notation, such as the union and intersection properties