Which Graph Shows The Solution Set For − 5 2 X − 3 ≤ 2 -\frac{5}{2} X - 3 \leq 2 − 2 5 ​ X − 3 ≤ 2 ?

Introduction

In mathematics, graphing inequalities is a crucial concept that helps us visualize the solution sets of various equations. When dealing with linear inequalities, we can represent them graphically on a coordinate plane. In this article, we will explore how to graph the solution set for the inequality $-\frac{5}{2} x - 3 \leq 2$. We will analyze the inequality, identify the key components, and determine which graph represents the solution set.

Understanding the Inequality

The given inequality is $-\frac{5}{2} x - 3 \leq 2$. To begin, we need to isolate the variable $x$ by performing algebraic operations. First, let's add $3$ to both sides of the inequality:

$$-\frac{5}{2} x - 3 + 3 \leq 2 + 3$$

This simplifies to:

$$-\frac{5}{2} x \leq 5$$

Next, we multiply both sides by $-\frac{2}{5}$ to solve for $x$. However, since we are multiplying by a negative number, we need to reverse the direction of the inequality:

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

This simplifies to:

$$x \geq -2$$

Graphing the Solution Set

To graph the solution set, we need to represent the inequality $x \geq -2$ on a coordinate plane. The solution set consists of all points that satisfy the inequality. Since the inequality is in the form $x \geq a$, where $a$ is a constant, the solution set is a half-line that extends to the right of the point $(-2, 0)$.

Analyzing the Graphs

Let's analyze the possible graphs that could represent the solution set for the inequality $x \geq -2$. We will consider the following graphs:

• Graph A: A line passing through the point $(-2, 0)$ and extending to the right.
• Graph B: A line passing through the point $(-2, 0)$ and extending to the left.
• Graph C: A line passing through the point $(-2, 0)$ and extending to the right, but with a gap at the point $(-2, 0)$.
• Graph D: A line passing through the point $(-2, 0)$ and extending to the right, but with a gap at the point $(-2, 0)$ and a second line extending to the left.

Conclusion

Based on our analysis, the correct graph that represents the solution set for the inequality $x \geq -2$ is Graph A. This graph accurately represents the half-line that extends to the right of the point $(-2, 0)$.

Final Answer

The final answer is Graph A.

Additional Information

In conclusion, graphing inequalities is a crucial concept in mathematics that helps us visualize the solution sets of various equations. By analyzing the inequality $-\frac{5}{2} x - 3 \leq 2$, we were able to identify the key components and determine which graph represents the solution set. The correct graph is Graph A, which accurately represents the half-line that extends to the right of the point $(-2, 0)$.

Introduction

Graphing inequalities is a fundamental concept in mathematics that helps us visualize the solution sets of various equations. In our previous article, we explored how to graph the solution set for the inequality $-\frac{5}{2} x - 3 \leq 2$. In this article, we will address some frequently asked questions (FAQs) about graphing inequalities.

Q&A

Q1: What is the difference between graphing an equation and graphing an inequality?

A1: Graphing an equation represents a specific set of points that satisfy the equation, whereas graphing an inequality represents a range of values that satisfy the inequality.

Q2: How do I determine the direction of the inequality when graphing?

A2: When graphing an inequality, you need to determine the direction of the inequality by considering the sign of the coefficient of the variable. If the coefficient is positive, the inequality is directed to the right. If the coefficient is negative, the inequality is directed to the left.

Q3: What is the significance of the boundary line in graphing inequalities?

A3: The boundary line in graphing inequalities represents the point at which the inequality changes direction. It is essential to include the boundary line in the graph to accurately represent the solution set.

Q4: How do I graph an inequality with a variable on both sides?

A4: To graph an inequality with a variable on both sides, you need to isolate the variable on one side of the inequality. Then, you can graph the inequality as usual.

Q5: Can I graph an inequality with a fraction as the coefficient?

A5: Yes, you can graph an inequality with a fraction as the coefficient. To do this, you need to multiply both sides of the inequality by the reciprocal of the fraction to eliminate the fraction.

Q6: How do I graph an inequality with a negative coefficient?

A6: To graph an inequality with a negative coefficient, you need to reverse the direction of the inequality. For example, if the inequality is $x < 2$, you would graph the inequality as $x > -2$.

Q7: Can I graph an inequality with a zero on the right-hand side?

A7: Yes, you can graph an inequality with a zero on the right-hand side. In this case, the inequality is represented by a vertical line passing through the point $(0, 0)$.

Q8: How do I graph an inequality with a negative number on the right-hand side?

A8: To graph an inequality with a negative number on the right-hand side, you need to reverse the direction of the inequality. For example, if the inequality is $x > -2$, you would graph the inequality as $x < 2$.

Conclusion

Graphing inequalities is a fundamental concept in mathematics that helps us visualize the solution sets of various equations. By understanding the basics of graphing inequalities, you can accurately represent the solution sets of various equations. In this article, we addressed some frequently asked questions (FAQs) about graphing inequalities, providing you with a better understanding of this essential concept.

Additional Information

In conclusion, graphing inequalities is a crucial concept in mathematics that helps us visualize the solution sets of various equations. By understanding the basics of graphing inequalities, you can accurately represent the solution sets of various equations. We hope this article has provided you with a better understanding of graphing inequalities and has helped you to address any questions or concerns you may have had.