Solve The Inequality: $6v + 5 \ \textgreater \ -2v - 3$10. Solve The Inequality: Y + 4.5 \textless 10 Y + 4.5 \ \textless \ 10 Y + 4.5 \textless 10 Write An Equation For Each Of The Lines Described:11. A Line That Passes Through The Point ( 1 , − 2 (1, -2 ( 1 , − 2 ] And Is

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Introduction

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In mathematics, inequalities and equations are used to represent relationships between variables. Inequalities are used to compare values, while equations are used to represent equal values. In this article, we will focus on solving inequalities and writing equations for lines.

Solving Inequalities

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Solving the Inequality: $6v + 5 \ \textgreater \ -2v - 3$

To solve the inequality $6v + 5 \ \textgreater \ -2v - 3$, we need to isolate the variable $v$ on one side of the inequality.

First, let's add $2v$ to both sides of the inequality to get:

$$6v + 2v + 5 \ \textgreater \ -2v + 2v - 3$$

This simplifies to:

$$8v + 5 \ \textgreater \ -3$$

Next, let's subtract $5$ from both sides of the inequality to get:

$$8v + 5 - 5 \ \textgreater \ -3 - 5$$

This simplifies to:

$$8v \ \textgreater \ -8$$

Finally, let's divide both sides of the inequality by $8$ to get:

$$\frac{8v}{8} \ \textgreater \ \frac{-8}{8}$$

This simplifies to:

$$v \ \textgreater \ -1$$

Therefore, the solution to the inequality $6v + 5 \ \textgreater \ -2v - 3$ is $v \ \textgreater \ -1$.

Solving the Inequality: $y + 4.5 \ \textless \ 10$

To solve the inequality $y + 4.5 \ \textless \ 10$, we need to isolate the variable $y$ on one side of the inequality.

Let's subtract $4.5$ from both sides of the inequality to get:

$$y + 4.5 - 4.5 \ \textless \ 10 - 4.5$$

This simplifies to:

$$y \ \textless \ 5.5$$

Therefore, the solution to the inequality $y + 4.5 \ \textless \ 10$ is $y \ \textless \ 5.5$.

Writing Equations for Lines

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Writing an Equation for a Line that Passes Through the Point $(1, -2)$

To write an equation for a line that passes through the point $(1, -2)$, we need to use the point-slope form of a linear equation, which is:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is the point through which the line passes, and $m$ is the slope of the line.

Let's use the point $(1, -2)$ as the point through which the line passes. We can choose any value for the slope $m$. Let's choose $m = 2$.

Plugging in the values, we get:

$$y - (-2) = 2(x - 1)$$

This simplifies to:

$$y + 2 = 2x - 2$$

Finally, let's add $2$ to both sides of the equation to get:

$$y = 2x - 4$$

Therefore, the equation for the line that passes through the point $(1, -2)$ is $y = 2x - 4$.

Conclusion

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In this article, we have solved two inequalities and written an equation for a line that passes through a given point. We have used the point-slope form of a linear equation to write the equation for the line.

Key Takeaways

• To solve an inequality, we need to isolate the variable on one side of the inequality.
• To write an equation for a line that passes through a given point, we can use the point-slope form of a linear equation.
• The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point through which the line passes, and $m$ is the slope of the line.

Future Directions

• We can use the equations we have written to graph the lines and visualize the relationships between the variables.
• We can also use the equations to solve systems of linear equations and inequalities.
• We can explore other types of equations, such as quadratic equations and polynomial equations, and learn how to solve them.

References

• Math Open Reference
• Khan Academy
• Mathway

Related Topics

• Linear Equations
• Graphing Lines
• Systems of Linear Equations
• Quadratic Equations
• Polynomial Equations
  

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Introduction

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In our previous article, we discussed solving inequalities and writing equations for lines. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topics.

Q&A

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Q: What is the difference between an inequality and an equation?

A: An inequality is a statement that compares two values, indicating whether one value is greater than, less than, or equal to another value. An equation, on the other hand, is a statement that expresses the equality of two values.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point through which the line passes, and $m$ is the slope of the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the equation of a line that passes through the point $(1, -2)$ and has a slope of $2$?

A: The equation of the line is $y = 2x - 4$.

Q: How do I graph a line?

A: To graph a line, you can use the equation of the line to find the x and y intercepts, and then plot the points on a coordinate plane.

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

A: A linear equation is an equation that can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants. A quadratic equation, on the other hand, is an equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the difference between a polynomial equation and a rational equation?

A: A polynomial equation is an equation that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, where $a_n$, $a_{n-1}$, $\ldots$, $a_1$, and $a_0$ are constants. A rational equation, on the other hand, is an equation that can be written in the form $\frac{p(x)}{q(x)} = 0$, where $p(x)$ and $q(x)$ are polynomials.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can multiply both sides of the equation by the denominator to eliminate the fraction, and then solve the resulting equation.

Conclusion

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In this article, we have provided a Q&A section to help clarify any doubts and provide additional information on the topics of solving inequalities and writing equations for lines. We hope that this article has been helpful in providing a better understanding of these topics.

Key Takeaways

• To solve an inequality, you need to isolate the variable on one side of the inequality.
• The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point through which the line passes, and $m$ is the slope of the line.
• To find the slope of a line, you can use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
• To graph a line, you can use the equation of the line to find the x and y intercepts, and then plot the points on a coordinate plane.

Future Directions

• We can use the equations we have written to graph the lines and visualize the relationships between the variables.
• We can also use the equations to solve systems of linear equations and inequalities.
• We can explore other types of equations, such as quadratic equations and polynomial equations, and learn how to solve them.

References

• Math Open Reference
• Khan Academy
• Mathway

Related Topics

• Linear Equations
• Graphing Lines
• Systems of Linear Equations
• Quadratic Equations
• Polynomial Equations