Solve The Inequality For { U $} . . . { -\frac{10}{3} U \ \textless \ 6 \} Simplify Your Answer As Much As Possible.

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Introduction

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In this article, we will focus on solving the given inequality for the variable u. The inequality is represented as $-\frac{10}{3}u < 6$. Our goal is to isolate the variable u and simplify the expression as much as possible.

Understanding the Inequality

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Before we begin solving the inequality, it's essential to understand the concept of inequalities. An inequality is a statement that compares two expressions using a mathematical symbol, such as <, >, ≤, or ≥. In this case, we have a less-than inequality, which means that the expression on the left-hand side is less than the expression on the right-hand side.

Step 1: Isolate the Variable u

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To solve the inequality, we need to isolate the variable u. This means that we need to get u by itself on one side of the inequality. To do this, we can start by multiplying both sides of the inequality by -3, which is the reciprocal of -10/3.

-\frac{10}{3}u < 6
\Rightarrow u > -\frac{3}{5} \cdot 6
\Rightarrow u > -\frac{18}{5}

Step 2: Simplify the Expression

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Now that we have isolated the variable u, we can simplify the expression. To do this, we can divide both sides of the inequality by -10/3.

u > -\frac{18}{5}
\Rightarrow u > -\frac{18}{5}

Step 3: Write the Solution in Interval Notation

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The solution to the inequality can be written in interval notation as $(-\infty, -\frac{18}{5})$. This means that u is less than -18/5.

Conclusion

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In conclusion, we have solved the inequality for the variable u. The solution is $u > -\frac{18}{5}$. This means that u is greater than -18/5.

Frequently Asked Questions

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Q: What is the solution to the inequality?

A: The solution to the inequality is $u > -\frac{18}{5}$.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality and simplify the expression.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using parentheses and brackets.

Final Answer

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The final answer is $\boxed{u > -\frac{18}{5}}$.

Additional Resources

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For more information on solving inequalities, please refer to the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Related Topics

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• Solving Linear Equations
• Solving Quadratic Equations
• Graphing Inequalities

Keywords

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• Solving Inequalities
• Isolating Variables
• Simplifying Expressions
• Interval Notation
• Linear Equations
• Quadratic Equations
• Graphing Inequalities
  

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Introduction

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In our previous article, we discussed how to solve the inequality $-\frac{10}{3}u < 6$. In this article, we will provide a Q&A guide to help you understand the concept of solving inequalities and how to apply it to different types of inequalities.

Q&A

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Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a mathematical symbol, such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality and simplify the expression.

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

A: A linear inequality is an inequality that can be written in the form $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can add or subtract the same value to both sides of the inequality to isolate the variable.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can factor the quadratic expression and then use the sign of the expression to determine the solution.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using parentheses and brackets.

Q: How do I write the solution to an inequality in interval notation?

A: To write the solution to an inequality in interval notation, you need to determine the values of the variable that satisfy the inequality and then use parentheses and brackets to represent the solution.

Q: What is the difference between a closed interval and an open interval?

A: A closed interval is an interval that includes the endpoints, while an open interval is an interval that does not include the endpoints.

Q: How do I determine the solution to an inequality using a number line?

A: To determine the solution to an inequality using a number line, you need to plot the values of the variable on the number line and then determine the values that satisfy the inequality.

Examples

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Example 1: Solving a Linear Inequality

Solve the inequality $2x + 3 < 5$.

2x + 3 < 5
\Rightarrow 2x < 2
\Rightarrow x < 1

Example 2: Solving a Quadratic Inequality

Solve the inequality $x^2 + 4x + 4 > 0$.

x^2 + 4x + 4 > 0
\Rightarrow (x + 2)^2 > 0
\Rightarrow x + 2 > 0
\Rightarrow x > -2

Example 3: Writing the Solution to an Inequality in Interval Notation

Write the solution to the inequality $x < 2$ in interval notation.

x < 2
\Rightarrow (-\infty, 2)

Conclusion

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In conclusion, solving inequalities is an important concept in mathematics that can be applied to a wide range of problems. By understanding how to solve linear and quadratic inequalities, you can determine the solution to a variety of inequalities and write the solution in interval notation.

Frequently Asked Questions

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

A: A linear inequality is an inequality that can be written in the form $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can add or subtract the same value to both sides of the inequality to isolate the variable.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can factor the quadratic expression and then use the sign of the expression to determine the solution.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using parentheses and brackets.

Final Answer

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The final answer is $\boxed{(-\infty, 2)}$.

Additional Resources

---

For more information on solving inequalities, please refer to the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Related Topics

---

• Solving Linear Equations
• Solving Quadratic Equations
• Graphing Inequalities

Keywords

---

• Solving Inequalities
• Linear Inequalities
• Quadratic Inequalities
• Interval Notation
• Number Line
• Closed Intervals
• Open Intervals