Solve The Following Inequality For Bb. Write Your Answer In Simplest Form. 5, Plus, 2, Left Parenthesis, 6, B, Minus, 10, Right Parenthesis, Is Greater Than, Minus, B, Plus, 8, Minus, 7 5+2(6b−10)> −b+8−7

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Introduction

In this article, we will focus on solving linear inequalities, specifically the given inequality 5 + 2(6b - 10) > -b + 8 - 7. Linear inequalities are a fundamental concept in mathematics, and solving them requires a clear understanding of algebraic operations and properties. In this discussion, we will break down the solution process into manageable steps, making it easier for readers to grasp the concept.

Understanding the Inequality

The given inequality is 5 + 2(6b - 10) > -b + 8 - 7. To solve this inequality, we need to isolate the variable b on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality by evaluating the expression inside the parentheses.

Simplifying the Left-Hand Side

The left-hand side of the inequality is 5 + 2(6b - 10). To simplify this expression, we need to multiply 2 by the terms inside the parentheses.

# Simplifying the left-hand side
left_hand_side = 5 + 2 * (6 * b - 10)

Using the distributive property, we can rewrite the expression as:

5 + 12b - 20

Now, let's combine like terms:

-5 + 12b

So, the simplified left-hand side of the inequality is -5 + 12b.

Simplifying the Right-Hand Side

The right-hand side of the inequality is -b + 8 - 7. To simplify this expression, we need to combine like terms.

# Simplifying the right-hand side
right_hand_side = -b + 8 - 7

Using the commutative property, we can rewrite the expression as:

-b + 1

Now, let's combine like terms:

-b + 1

So, the simplified right-hand side of the inequality is -b + 1.

Combining Like Terms

Now that we have simplified both sides of the inequality, we can combine like terms.

# Combining like terms
simplified_inequality = -5 + 12b > -b + 1

Isolating the Variable b

To isolate the variable b, we need to get all the terms with b on one side of the inequality sign. Let's start by adding b to both sides of the inequality.

# Adding b to both sides
inequality = -5 + 13b > 1

Next, let's add 5 to both sides of the inequality.

# Adding 5 to both sides
inequality = 13b > 6

Solving for b

Now that we have isolated the variable b, we can solve for b by dividing both sides of the inequality by 13.

# Dividing both sides by 13
solution = b > 6/13

Therefore, the solution to the inequality is b > 6/13.

Conclusion

In this article, we solved the linear inequality 5 + 2(6b - 10) > -b + 8 - 7 by simplifying both sides of the inequality, combining like terms, and isolating the variable b. We then solved for b by dividing both sides of the inequality by 13. The solution to the inequality is b > 6/13. This example demonstrates the importance of following the order of operations and using algebraic properties to solve linear inequalities.

Final Answer

Introduction

In our previous article, we solved the linear inequality 5 + 2(6b - 10) > -b + 8 - 7 by simplifying both sides of the inequality, combining like terms, and isolating the variable b. In this article, we will provide a Q&A guide to help readers understand the concept of solving linear inequalities.

Q: What is a linear 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, and x is the variable.

Q: How do I simplify a linear inequality?

A: To simplify a linear inequality, you need to follow the order of operations (PEMDAS):

  1. Evaluate expressions inside parentheses.
  2. Exponentiate (if necessary).
  3. Multiply and divide from left to right.
  4. Add and subtract from left to right.

Q: How do I combine like terms in a linear inequality?

A: To combine like terms in a linear inequality, you need to add or subtract the coefficients of the like terms.

Q: How do I isolate the variable in a linear inequality?

A: To isolate the variable in a linear inequality, you need to get all the terms with the variable on one side of the inequality sign. You can do this by adding or subtracting the same value to both sides of the inequality.

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

A: A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ax + b > c, where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear inequality with fractions?

A: To solve a linear inequality with fractions, you need to follow the same steps as solving a linear inequality with whole numbers. However, you may need to multiply both sides of the inequality by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: Can I use the same methods to solve quadratic inequalities?

A: No, quadratic inequalities are more complex and require different methods to solve. Quadratic inequalities can be written in the form ax^2 + bx + c > d, where a, b, c, and d are constants, and x is the variable.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include:

  • Not following the order of operations (PEMDAS)
  • Not combining like terms
  • Not isolating the variable
  • Not checking the direction of the inequality sign

Conclusion

In this article, we provided a Q&A guide to help readers understand the concept of solving linear inequalities. We covered topics such as simplifying linear inequalities, combining like terms, isolating the variable, and avoiding common mistakes. By following these steps and avoiding common mistakes, readers can become proficient in solving linear inequalities.

Final Tips

  • Practice solving linear inequalities with different coefficients and variables.
  • Use online resources or textbooks to supplement your learning.
  • Join a study group or ask a tutor for help if you need additional support.

Common Linear Inequality Examples

  • 2x + 5 > 3
  • x - 2 < 4
  • 3x + 1 > 2x - 3

Solving Linear Inequalities: A Practice Set

  1. Solve the inequality 2x + 3 > 5.
  2. Solve the inequality x - 2 < 3.
  3. Solve the inequality 3x + 2 > 2x - 1.

Answer Key

  1. x > 1
  2. x < 5
  3. x > -1/3