Solve The Linear Inequality For $y$. Graph Your Answer On The Number Line. Then Write The Solution In Inequality Notation. Round Your Answers To The Nearest Hundredth.$-15.5y \ \textless \ -54.25$Show Your Work Here:Hint: To Add

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Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities for the variable yy, graphing the solution on a number line, and writing the solution in inequality notation. We will use the given inequality −15.5y<−54.25-15.5y < -54.25 as an example and provide a step-by-step guide on how to solve it.

Step 1: Isolate the Variable

The first step in solving a linear inequality is to isolate the variable yy on one side of the inequality sign. To do this, we need to get rid of the coefficient of yy, which is −15.5-15.5. We can do this by dividing both sides of the inequality by −15.5-15.5. However, when we divide or multiply both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

-15.5y < -54.25
\Rightarrow \frac{-15.5y}{-15.5} > \frac{-54.25}{-15.5}
\Rightarrow y > \frac{54.25}{15.5}

Step 2: Simplify the Expression

Now that we have isolated the variable yy, we need to simplify the expression on the right-hand side of the inequality. We can do this by dividing 54.2554.25 by 15.515.5.

y > \frac{54.25}{15.5}
\Rightarrow y > 3.49

Step 3: Graph the Solution on a Number Line

To graph the solution on a number line, we need to plot a point on the number line that represents the value of yy that satisfies the inequality. In this case, we can plot a point at y=3.49y = 3.49. Since the inequality is greater than, we need to shade the region to the right of the point.

Step 4: Write the Solution in Inequality Notation

Finally, we need to write the solution in inequality notation. Since the inequality is greater than, we can write it as y>3.49y > 3.49.

Conclusion

In this article, we have shown how to solve a linear inequality for the variable yy, graph the solution on a number line, and write the solution in inequality notation. We have used the given inequality −15.5y<−54.25-15.5y < -54.25 as an example and provided a step-by-step guide on how to solve it. By following these steps, students can master the skill of solving linear inequalities and apply it to a wide range of mathematical problems.

Additional Examples

Here are a few additional examples of linear inequalities that can be solved using the same steps:

  • 2.5y<10.252.5y < 10.25
  • −3.2y>−12.8-3.2y > -12.8
  • 1.8y≥9.21.8y \geq 9.2

Tips and Tricks

Here are a few tips and tricks that can help students solve linear inequalities:

  • Always isolate the variable on one side of the inequality sign.
  • When dividing or multiplying both sides of an inequality by a negative number, reverse the direction of the inequality sign.
  • Simplify the expression on the right-hand side of the inequality.
  • Graph the solution on a number line and shade the region that satisfies the inequality.
  • Write the solution in inequality notation.

Common Mistakes

Here are a few common mistakes that students make when solving linear inequalities:

  • Failing to isolate the variable on one side of the inequality sign.
  • Not reversing the direction of the inequality sign when dividing or multiplying both sides by a negative number.
  • Not simplifying the expression on the right-hand side of the inequality.
  • Not graphing the solution on a number line and shading the region that satisfies the inequality.
  • Not writing the solution in inequality notation.

Conclusion

Introduction

Solving linear inequalities can be a challenging task for many students. However, with the right approach and practice, it can become a manageable and even enjoyable process. In this article, we will provide a Q&A guide to help students understand and solve linear inequalities.

Q: What is a linear inequality?

A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form of ax + b, where a and b are constants and x is the variable.

A: For example, the inequality 2x + 3 > 5 is a linear inequality because it involves a linear expression (2x + 3) and a constant (5).

Q: How do I solve a linear inequality?

To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

A: For example, to solve the inequality 2x + 3 > 5, you can subtract 3 from both sides to get 2x > 2, and then divide both sides by 2 to get x > 1.

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

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

A: For example, the equation 2x + 3 = 5 is a linear equation because it involves a linear expression (2x + 3) and a constant (5), and is equal to zero. The inequality 2x + 3 > 5, on the other hand, is a linear inequality because it involves a linear expression (2x + 3) and a constant (5), and is not equal to zero.

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

To graph a linear inequality on a number line, you need to plot a point on the number line that represents the value of the variable that satisfies the inequality. If the inequality is greater than or equal to, you need to shade the region to the right of the point. If the inequality is less than or equal to, you need to shade the region to the left of the point.

A: For example, to graph the inequality x > 1 on a number line, you can plot a point at x = 1 and shade the region to the right of the point.

Q: What is the solution to a linear inequality?

The solution to a linear inequality is the set of all values of the variable that satisfy the inequality.

A: For example, the solution to the inequality x > 1 is the set of all values of x that are greater than 1.

Q: How do I write the solution to a linear inequality in inequality notation?

To write the solution to a linear inequality in inequality notation, you need to use the following format:

  • If the inequality is greater than, use the symbol >.
  • If the inequality is less than, use the symbol <.
  • If the inequality is greater than or equal to, use the symbol ≥.
  • If the inequality is less than or equal to, use the symbol ≤.

A: For example, the solution to the inequality x > 1 can be written in inequality notation as x > 1.

Conclusion

Solving linear inequalities can be a challenging task, but with practice and the right approach, it can become manageable and even enjoyable. By following the steps outlined in this article, students can solve linear inequalities and apply it to a wide range of mathematical problems.

Additional Resources

Here are a few additional resources that can help students learn more about solving linear inequalities:

  • Khan Academy: Solving Linear Inequalities
  • Mathway: Solving Linear Inequalities
  • IXL: Solving Linear Inequalities

Practice Problems

Here are a few practice problems that can help students practice solving linear inequalities:

  • Solve the inequality 2x + 3 > 5.
  • Solve the inequality x - 2 < 3.
  • Solve the inequality 3x + 2 ≥ 7.
  • Solve the inequality x - 1 ≤ 2.

Answer Key

Here are the answers to the practice problems:

  • 2x + 3 > 5 → x > 1
  • x - 2 < 3 → x < 5
  • 3x + 2 ≥ 7 → x ≥ 2
  • x - 1 ≤ 2 → x ≤ 3