3-2(u - 1) > 8 + U; 2) 5(u+2)+14 < 6 U; (3+8u) > 6,25+u; 3) 11 (3+ 4) 4(u+3) < 3(u + 2); 5) 3(2u+1)5(u-1); 6) 35 5u - 23 3 ​

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In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. In this article, we will focus on solving six inequalities, each with its unique characteristics and requirements. We will break down each inequality, provide step-by-step solutions, and offer explanations to help you understand the underlying concepts.

Inequality 1: 3-2(u - 1) > 8 + u

The first inequality we will tackle is 3-2(u - 1) > 8 + u. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Step 1: Distribute the negative sign

The first step is to distribute the negative sign to the terms inside the parentheses: -2(u - 1) = -2u + 2.

Step 2: Simplify the expression

Now, we can simplify the expression by combining like terms: 3 - 2u + 2 > 8 + u.

Step 3: Combine like terms

Combine the constant terms: 5 - 2u > 8 + u.

Step 4: Move all terms to one side

Subtract u from both sides: 5 - 3u > 8.

Step 5: Add 3u to both sides

Add 3u to both sides: 5 > 8 + 3u.

Step 6: Subtract 8 from both sides

Subtract 8 from both sides: -3 > 3u.

Step 7: Divide both sides by 3

Divide both sides by 3: -1 > u.

The final solution to the first inequality is u < -1.

Inequality 2: 5(u+2)+14 < 6 u

The second inequality we will tackle is 5(u+2)+14 < 6 u. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Step 1: Distribute the 5

The first step is to distribute the 5 to the terms inside the parentheses: 5u + 10 + 14 < 6u.

Step 2: Combine like terms

Combine the constant terms: 5u + 24 < 6u.

Step 3: Subtract 5u from both sides

Subtract 5u from both sides: 24 < u.

The final solution to the second inequality is u > 24.

Inequality 3: (3+8u) > 6,25+u

The third inequality we will tackle is (3+8u) > 6,25+u. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Step 1: Distribute the 8

The first step is to distribute the 8 to the terms inside the parentheses: 3 + 8u > 6.25 + u.

Step 2: Subtract u from both sides

Subtract u from both sides: 3 + 7u > 6.25.

Step 3: Subtract 3 from both sides

Subtract 3 from both sides: 7u > 3.25.

Step 4: Divide both sides by 7

Divide both sides by 7: u > 0.464.

The final solution to the third inequality is u > 0.464.

Inequality 4: 11 (3+ 4) 4(u+3) < 3(u + 2)

The fourth inequality we will tackle is 11 (3+ 4) 4(u+3) < 3(u + 2). To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Step 1: Distribute the 11

The first step is to distribute the 11 to the terms inside the parentheses: 11(3 + 4)4(u + 3) < 3(u + 2).

Step 2: Simplify the expression

Now, we can simplify the expression by combining like terms: 11(7)4(u + 3) < 3(u + 2).

Step 3: Multiply the numbers

Multiply the numbers: 308(u + 3) < 3(u + 2).

Step 4: Distribute the 308

Distribute the 308 to the terms inside the parentheses: 308u + 924 < 3u + 6.

Step 5: Subtract 3u from both sides

Subtract 3u from both sides: 305u + 924 < 6.

Step 6: Subtract 924 from both sides

Subtract 924 from both sides: 305u < -918.

Step 7: Divide both sides by 305

Divide both sides by 305: u < -3.

The final solution to the fourth inequality is u < -3.

Inequality 5: 3(2u+1)5(u-1)

The fifth inequality we will tackle is 3(2u+1)5(u-1). To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Step 1: Distribute the 3

The first step is to distribute the 3 to the terms inside the parentheses: 3(2u + 1)5(u - 1).

Step 2: Simplify the expression

Now, we can simplify the expression by combining like terms: 6u + 3)5(u - 1).

Step 3: Multiply the numbers

Multiply the numbers: (6u + 3)(5u - 1).

Step 4: Distribute the (6u + 3)

Distribute the (6u + 3) to the terms inside the parentheses: 30u^2 - 6u + 15u - 3.

Step 5: Combine like terms

Combine the like terms: 30u^2 + 9u - 3.

The final solution to the fifth inequality is 30u^2 + 9u - 3.

Inequality 6: 35 5u - 23 3

The sixth inequality we will tackle is 35 5u - 23 3. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Step 1: Distribute the 35

The first step is to distribute the 35 to the terms inside the parentheses: 35(5u) - 23(3).

Step 2: Simplify the expression

Now, we can simplify the expression by combining like terms: 175u - 69.

The final solution to the sixth inequality is 175u - 69.

In conclusion, solving inequalities requires a step-by-step approach, and each inequality has its unique characteristics and requirements. By following the order of operations (PEMDAS) and simplifying the expression, we can solve each inequality and find the final solution.

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In the previous article, we solved six inequalities, each with its unique characteristics and requirements. In this article, we will provide a Q&A section to help you better understand the concepts and solutions.

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

A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. An equation, on the other hand, is a statement that says two expressions are equal.

Q: How do I know which inequality sign to use?

A: The inequality sign depends on the direction of the inequality. If the expression on the left is greater than the expression on the right, use the "greater than" sign (>). If the expression on the left is less than the expression on the right, use the "less than" sign (<).

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an inequality expression?

A: To simplify an inequality expression, follow these steps:

  1. Distribute any coefficients to the terms inside the parentheses.
  2. Combine like terms.
  3. Move all terms to one side of the inequality sign.
  4. 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, where a, b, and c are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form ax^2 + bx + c < d, where a, b, c, and d are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, follow these steps:

  1. Factor the quadratic expression, if possible.
  2. Set each factor equal to zero and solve for x.
  3. Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the final solution to the inequalities we solved?

A: The final solutions to the inequalities we solved are:

  1. u < -1
  2. u > 24
  3. u > 0.464
  4. u < -3
  5. 30u^2 + 9u - 3
  6. 175u - 69

We hope this Q&A section has helped you better understand the concepts and solutions to the inequalities we solved. If you have any further questions, please don't hesitate to ask.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not following the order of operations (PEMDAS)
  • Not distributing coefficients to terms inside parentheses
  • Not combining like terms
  • Not moving all terms to one side of the inequality sign
  • Not simplifying the expression

By avoiding these common mistakes, you can ensure that your solutions are accurate and complete.

Conclusion

Solving inequalities requires a step-by-step approach, and each inequality has its unique characteristics and requirements. By following the order of operations (PEMDAS) and simplifying the expression, you can solve each inequality and find the final solution. We hope this article has helped you better understand the concepts and solutions to the inequalities we solved. If you have any further questions, please don't hesitate to ask.