Rewrite The Inequality { -2x - 4y \geq 8$}$ In Slope-intercept Form.

Introduction

In mathematics, inequalities are used to describe relationships between variables. The slope-intercept form is a specific way of writing linear equations and inequalities, where the variable y is expressed in terms of the variable x and a constant term. In this article, we will focus on rewriting the inequality ${-2x - 4y \geq 8}$ in slope-intercept form.

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation or inequality is given by the equation ${y = mx + b}$, where m is the slope of the line and b is the y-intercept. The slope-intercept form is useful for graphing lines and understanding the relationship between the variables.

Rewriting the Inequality

To rewrite the inequality ${-2x - 4y \geq 8}$ in slope-intercept form, we need to isolate the variable y. We can do this by adding 4y to both sides of the inequality and then dividing both sides by -4.

Step 1: Add 4y to both sides of the inequality

${-2x - 4y \geq 8}$ ${4y \geq -2x - 8}$ ${4y + 2x \geq -8}$

Step 2: Divide both sides of the inequality by -4

${\frac{4y + 2x}{-4} \geq \frac{-8}{-4}}$ ${-y - \frac{1}{2}x \geq 2}$

Step 3: Multiply both sides of the inequality by -1 to make the coefficient of y positive

${-(-y - \frac{1}{2}x) \geq -2}$ ${y + \frac{1}{2}x \leq -2}$

Step 4: Rewrite the inequality in slope-intercept form

${y \leq -\frac{1}{2}x - 2}$

Conclusion

In this article, we have rewritten the inequality ${-2x - 4y \geq 8}$ in slope-intercept form. The final answer is ${y \leq -\frac{1}{2}x - 2}$. This form is useful for graphing lines and understanding the relationship between the variables.

Tips and Tricks

• When rewriting an inequality in slope-intercept form, make sure to isolate the variable y.
• Use the correct order of operations when simplifying the inequality.
• Check the direction of the inequality sign to ensure that it is correct.

Common Mistakes

• Failing to isolate the variable y.
• Not using the correct order of operations.
• Ignoring the direction of the inequality sign.

Real-World Applications

The slope-intercept form is used in a variety of real-world applications, including:

• Graphing lines and understanding the relationship between variables.
• Solving systems of linear equations and inequalities.
• Modeling real-world situations using linear equations and inequalities.

Practice Problems

1. Rewrite the inequality ${3x - 2y \geq 6}$ in slope-intercept form.
2. Rewrite the inequality ${-4x + 3y \leq 9}$ in slope-intercept form.
3. Rewrite the inequality ${2x + 5y \geq 15}$ in slope-intercept form.

Solutions

1. ${y \leq \frac{3}{2}x - 3}$
2. ${y \geq -\frac{4}{3}x + 3}$
3. ${y \geq -\frac{2}{5}x + 3}$

Conclusion

Q: What is the slope-intercept form of a linear equation or inequality?

A: The slope-intercept form of a linear equation or inequality is given by the equation ${y = mx + b}$, where m is the slope of the line and b is the y-intercept.

Q: How do I rewrite an inequality in slope-intercept form?

A: To rewrite an inequality in slope-intercept form, you need to isolate the variable y. You can do this by adding or subtracting terms from both sides of the inequality and then dividing both sides by the coefficient of y.

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

A: A linear equation is an equation that can be written in the form ${y = mx + b}$, where m is the slope of the line and b is the y-intercept. A linear inequality is an inequality that can be written in the form ${y \leq mx + b}$ or ${y \geq mx + b}$.

Q: How do I graph a linear equation or inequality in slope-intercept form?

A: To graph a linear equation or inequality in slope-intercept form, you can use the slope and y-intercept to find two points on the line. You can then draw a line through the two points to represent the equation or inequality.

Q: What are some common mistakes to avoid when rewriting an inequality in slope-intercept form?

A: Some common mistakes to avoid when rewriting an inequality in slope-intercept form include:

• Failing to isolate the variable y.
• Not using the correct order of operations.
• Ignoring the direction of the inequality sign.

Q: How do I check my work when rewriting an inequality in slope-intercept form?

A: To check your work when rewriting an inequality in slope-intercept form, you can plug in a test value for x and y to see if the inequality is true. You can also graph the inequality to see if it matches the original inequality.

Q: What are some real-world applications of rewriting inequalities in slope-intercept form?

A: Some real-world applications of rewriting inequalities in slope-intercept form include:

• Graphing lines and understanding the relationship between variables.
• Solving systems of linear equations and inequalities.
• Modeling real-world situations using linear equations and inequalities.

Q: How do I practice rewriting inequalities in slope-intercept form?

A: You can practice rewriting inequalities in slope-intercept form by working through practice problems and checking your work. You can also use online resources and math software to help you practice.

Q: What are some tips for rewriting inequalities in slope-intercept form?

A: Some tips for rewriting inequalities in slope-intercept form include:

• Make sure to isolate the variable y.
• Use the correct order of operations.
• Check the direction of the inequality sign.

Q: How do I know if I have rewritten an inequality correctly in slope-intercept form?

A: You can check if you have rewritten an inequality correctly in slope-intercept form by plugging in a test value for x and y to see if the inequality is true. You can also graph the inequality to see if it matches the original inequality.

Q: What are some common mistakes to avoid when graphing a linear equation or inequality in slope-intercept form?

A: Some common mistakes to avoid when graphing a linear equation or inequality in slope-intercept form include:

• Failing to use the correct slope and y-intercept.
• Not drawing a line through the two points.
• Ignoring the direction of the inequality sign.

Q: How do I check my work when graphing a linear equation or inequality in slope-intercept form?

A: To check your work when graphing a linear equation or inequality in slope-intercept form, you can plug in a test value for x and y to see if the inequality is true. You can also use online resources and math software to help you check your work.

Conclusion

In conclusion, rewriting inequalities in slope-intercept form is an important skill in mathematics. By following the steps outlined in this article and practicing regularly, you can become proficient in rewriting inequalities in slope-intercept form. Remember to isolate the variable y, use the correct order of operations, and check the direction of the inequality sign. With practice, you will become proficient in rewriting inequalities in slope-intercept form.