Solve Each Inequality And Graph The Solution.2) 3 ( 1 + 8 B ) ≤ B + 3 3(1+8b) \leq B+3 3 ( 1 + 8 B ) ≤ B + 3

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Introduction

In this article, we will focus on solving and graphing the solution to the given inequality: 3(1+8b)≤b+33(1+8b) \leq b+3. This type of problem is commonly encountered in algebra and mathematics, and requires a clear understanding of the concepts of inequalities and graphing.

Understanding the Inequality

The given inequality is 3(1+8b)≤b+33(1+8b) \leq b+3. To solve this inequality, we need to isolate the variable bb on one side of the inequality sign. The first step is to distribute the 3 on the left-hand side of the inequality.

Distributing the 3

When we distribute the 3 on the left-hand side of the inequality, we get:

3(1+8b)=3(1)+3(8b)3(1+8b) = 3(1) + 3(8b)

Using the distributive property, we can simplify this expression to:

3+24b3 + 24b

So, the inequality becomes:

3+24b≤b+33 + 24b \leq b + 3

Isolating the Variable

The next step is to isolate the variable bb on one side of the inequality sign. To do this, we need to get all the terms containing bb on one side of the inequality sign.

Subtracting 3 from Both Sides

Subtracting 3 from both sides of the inequality, we get:

3+24b−3≤b+3−33 + 24b - 3 \leq b + 3 - 3

Simplifying this expression, we get:

24b≤b24b \leq b

Subtracting b from Both Sides

Subtracting bb from both sides of the inequality, we get:

24b−b≤b−b24b - b \leq b - b

Simplifying this expression, we get:

23b≤023b \leq 0

Solving for b

The final step is to solve for bb. To do this, we need to isolate bb on one side of the inequality sign.

Dividing Both Sides by 23

Dividing both sides of the inequality by 23, we get:

23b23≤023\frac{23b}{23} \leq \frac{0}{23}

Simplifying this expression, we get:

b≤0b \leq 0

Graphing the Solution

The solution to the inequality is b≤0b \leq 0. To graph this solution, we need to draw a number line and mark the point where b=0b = 0.

Drawing the Number Line

A number line is a line that represents all the possible values of a variable. In this case, the variable is bb. We can draw a number line by marking the point where b=0b = 0.

Marking the Solution

The solution to the inequality is b≤0b \leq 0. To mark this solution on the number line, we need to shade the region to the left of the point where b=0b = 0.

Conclusion

In this article, we solved and graphed the solution to the given inequality: 3(1+8b)≤b+33(1+8b) \leq b+3. We used the distributive property to simplify the expression, and then isolated the variable bb on one side of the inequality sign. Finally, we graphed the solution on a number line.

Key Takeaways

  • To solve an inequality, we need to isolate the variable on one side of the inequality sign.
  • We can use the distributive property to simplify expressions.
  • To graph the solution to an inequality, we need to draw a number line and mark the point where the variable is equal to the solution.

Final Answer

The final answer is b≤0\boxed{b \leq 0}.

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Introduction

In the previous article, we solved and graphed the solution to the given inequality: 3(1+8b)≤b+33(1+8b) \leq b+3. In this article, we will answer some frequently asked questions related to solving and graphing inequalities.

Q&A

Q: What is an inequality?

A: An inequality is a statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate 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, or by multiplying or dividing both sides by the same non-zero value.

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≤cax + b \leq c, where aa, bb, and cc are constants. A quadratic inequality is an inequality that can be written in the form ax2+bx+c≤0ax^2 + bx + c \leq 0, where aa, bb, and cc are constants.

Q: How do I graph the solution to an inequality?

A: To graph the solution to an inequality, you need to draw a number line and mark the point where the variable is equal to the solution. Then, you need to shade the region to the left or right of the point, depending on whether the inequality is less than or greater than.

Q: What is the significance of the number line in graphing inequalities?

A: The number line is a visual representation of the possible values of a variable. It helps to identify the solution to an inequality by marking the point where the variable is equal to the solution and shading the region to the left or right of the point.

Q: Can I use a calculator to solve and graph inequalities?

A: Yes, you can use a calculator to solve and graph inequalities. However, it's always a good idea to check your work by hand to ensure that the solution is correct.

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

A: Some common mistakes to avoid when solving and graphing inequalities include:

  • Not isolating the variable on one side of the inequality sign
  • Not checking the direction of the inequality sign
  • Not shading the correct region on the number line
  • Not considering the restrictions on the variable

Conclusion

In this article, we answered some frequently asked questions related to solving and graphing inequalities. We covered topics such as the definition of an inequality, how to solve an inequality, and how to graph the solution to an inequality. We also discussed the significance of the number line in graphing inequalities and some common mistakes to avoid.

Key Takeaways

  • To solve an inequality, you need to isolate the variable on one side of the inequality sign.
  • To graph the solution to an inequality, you need to draw a number line and mark the point where the variable is equal to the solution.
  • The number line is a visual representation of the possible values of a variable.
  • It's always a good idea to check your work by hand to ensure that the solution is correct.

Final Answer

The final answer is Thereisnofinalnumericalanswertothisproblem.Thesolutionisasetofstepsandconceptstohelpyousolveandgraphinequalities.\boxed{There is no final numerical answer to this problem. The solution is a set of steps and concepts to help you solve and graph inequalities.}