Which Points Are Solutions To The Linear Inequality $y \ \textless \ 0.5x + 2$? Select Three Options.A. $(-3, -2$\]B. $(-2, 1$\]C. $(-1, -2$\]D. $(-1, 2$\]E. $(1, -2$\]

=====================================================

Introduction

---

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields such as economics, engineering, and computer science. In this article, we will focus on solving linear inequalities of the form $y < mx + b$, where $m$ and $b$ are constants. We will use the given inequality $y < 0.5x + 2$ as an example to illustrate the solution process.

Understanding Linear Inequalities

---

A linear inequality is an inequality that involves a linear expression. In the given inequality $y < 0.5x + 2$, the linear expression is $0.5x + 2$. The inequality states that the value of $y$ is less than the value of the linear expression.

Solving Linear Inequalities

---

To solve a linear inequality, we need to isolate the variable $y$ on one side of the inequality. In this case, we can do this by subtracting $0.5x + 2$ from both sides of the inequality.

Step 1: Subtract $0.5x + 2$ from both sides

$y < 0.5x + 2$

$y - 0.5x - 2 < 0$

Step 2: Factor out the coefficient of $x$

$y - 0.5x - 2 < 0$

$-0.5x - 2 < y$

Step 3: Add $0.5x$ to both sides

$-0.5x - 2 < y$

$-0.5x < y + 2$

Step 4: Add $2$ to both sides

$-0.5x < y + 2$

$-0.5x + 2 < y + 2 + 2$

$-0.5x + 2 < y + 4$

Step 5: Simplify the inequality

$-0.5x + 2 < y + 4$

$-0.5x < y + 4 - 2$

$-0.5x < y + 2$

Step 6: Divide both sides by $-0.5$

$-0.5x < y + 2$

$x > -\frac{y + 2}{0.5}$

$x > -2y - 4$

Finding the Solution Set

---

Now that we have isolated the variable $x$, we can find the solution set by substituting different values of $y$ into the inequality.

Substituting $y = 0$

$x > -2(0) - 4$

$x > -4$

Substituting $y = 1$

$x > -2(1) - 4$

$x > -6$

Substituting $y = -2$

$x > -2(-2) - 4$

$x > 0$

Graphing the Solution Set

---

To visualize the solution set, we can graph the inequality on a coordinate plane.

Graphing the inequality

The solution set is the region below the line $y = -0.5x + 2$.

Selecting the Correct Option

---

Now that we have found the solution set, we can select the correct option from the given choices.

Option A: $(-3, -2]$

This option is not correct because the solution set is the region below the line $y = -0.5x + 2$, and the interval $(-3, -2]$ is above the line.

Option B: $(-2, 1]$

This option is not correct because the solution set is the region below the line $y = -0.5x + 2$, and the interval $(-2, 1]$ is above the line.

Option C: $(-1, -2]$

This option is not correct because the solution set is the region below the line $y = -0.5x + 2$, and the interval $(-1, -2]$ is above the line.

Option D: $(-1, 2]$

This option is correct because the solution set is the region below the line $y = -0.5x + 2$, and the interval $(-1, 2]$ is below the line.

Option E: $(1, -2]$

This option is not correct because the solution set is the region below the line $y = -0.5x + 2$, and the interval $(1, -2]$ is above the line.

The final answer is: $\boxed{D}$

=====================================================

Introduction

---

In our previous article, we discussed how to solve linear inequalities of the form $y < mx + b$. We used the inequality $y < 0.5x + 2$ as an example to illustrate the solution process. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q&A

---

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression. In the inequality $y < 0.5x + 2$, the linear expression is $0.5x + 2$.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $y$ on one side of the inequality. You can do this by subtracting the linear expression from both sides of the inequality.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, and it is equal to a constant. A linear inequality, on the other hand, is an inequality that involves a linear expression, and it is not equal to a constant.

Q: Can I use the same methods to solve linear inequalities as I do to solve linear equations?

A: No, you cannot use the same methods to solve linear inequalities as you do to solve linear equations. Linear inequalities require a different approach, and you need to use techniques such as graphing and testing points to find the solution set.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality.

Q: What is the solution set of a linear inequality?

A: The solution set of a linear inequality is the set of all points that satisfy the inequality. It is the region below the line for inequalities of the form $y < mx + b$.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to be careful when using a calculator, as it may not always give you the correct solution.

Q: How do I check my solution to a linear inequality?

A: To check your solution to a linear inequality, you need to substitute the solution into the inequality and check if it is true.

Q: Can I use the same methods to solve systems of linear inequalities as I do to solve systems of linear equations?

A: No, you cannot use the same methods to solve systems of linear inequalities as you do to solve systems of linear equations. Systems of linear inequalities require a different approach, and you need to use techniques such as graphing and testing points to find the solution set.

Common Mistakes to Avoid

---

When solving linear inequalities, there are several common mistakes to avoid. Here are a few:

• Not isolating the variable: Make sure to isolate the variable $y$ on one side of the inequality.
• Not graphing the corresponding linear equation: Make sure to graph the corresponding linear equation to visualize the solution set.
• Not shading the correct region: Make sure to shade the region that satisfies the inequality.
• Not checking the solution: Make sure to check your solution to the inequality by substituting it into the inequality.

Conclusion

---

Solving linear inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to isolate the variable, graph the corresponding linear equation, and shade the correct region to find the solution set. By following these steps and avoiding common mistakes, you can solve linear inequalities with confidence.

Final Tips

---

Here are a few final tips to keep in mind when solving linear inequalities:

• Practice, practice, practice: The more you practice solving linear inequalities, the more comfortable you will become with the process.
• Use graphing to visualize the solution set: Graphing the corresponding linear equation can help you visualize the solution set and make it easier to find.
• Check your solution: Make sure to check your solution to the inequality by substituting it into the inequality.
• Use a calculator carefully: While calculators can be helpful in solving linear inequalities, be careful when using them, as they may not always give you the correct solution.

By following these tips and avoiding common mistakes, you can become proficient in solving linear inequalities and tackle even the most challenging problems with confidence.