Select The Correct Answer.Which Of The Following Graphs Shows The Solution Set For The Inequality $3|x+1|\ \textless \ 9$?A. $\[ \begin{array}{lllllllllllll} -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} \\]B.

Understanding the Inequality

The given inequality is $3|x+1| < 9$. To find the solution set, we need to isolate the absolute value expression and determine the range of values that satisfy the inequality.

Isolating the Absolute Value Expression

We can start by dividing both sides of the inequality by 3, which gives us $|x+1| < 3$. This step is valid because the absolute value of a number is always non-negative, and dividing by a positive number does not change the direction of the inequality.

Solving the Inequality

To solve the inequality, we need to consider two cases: when $x+1$ is positive and when $x+1$ is negative.

Case 1: $x+1 \geq 0$

When $x+1 \geq 0$, we can remove the absolute value bars and write the inequality as $x+1 < 3$. Subtracting 1 from both sides gives us $x < 2$.

Case 2: $x+1 < 0$

When $x+1 < 0$, we can remove the absolute value bars and write the inequality as $-(x+1) < 3$. Simplifying this expression gives us $-x-1 < 3$. Adding 1 to both sides gives us $-x < 4$. Multiplying both sides by -1 (which reverses the direction of the inequality) gives us $x > -4$.

Combining the Cases

We have found that the solution set for the inequality $3|x+1| < 9$ is $x < 2$ when $x+1 \geq 0$ and $x > -4$ when $x+1 < 0$. We can combine these two cases by writing the solution set as $x < 2$ or $x > -4$.

Graphing the Solution Set

To graph the solution set, we need to plot the lines $x=2$ and $x=-4$ on a number line. The solution set is the region between these two lines, excluding the points $x=2$ and $x=-4$.

Selecting the Correct Graph

Based on the graphing instructions, we can select the correct graph for the solution set of the inequality $3|x+1| < 9$.

Graph A

Graph A shows the solution set as the region between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$. This graph accurately represents the solution set of the inequality.

Graph B

Graph B does not accurately represent the solution set of the inequality. The graph shows the solution set as the region between the lines $x=2$ and $x=-4$, but it includes the points $x=2$ and $x=-4$.

Conclusion

Based on the graphing instructions, we can conclude that the correct graph for the solution set of the inequality $3|x+1| < 9$ is Graph A.

Final Answer

The final answer is Graph A.

Discussion

The solution set of the inequality $3|x+1| < 9$ is the region between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$. This can be represented graphically as a number line with the lines $x=2$ and $x=-4$ plotted on it. The solution set is the region between these two lines, excluding the points $x=2$ and $x=-4$.

Graph A

Graph A shows the solution set as the region between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$. This graph accurately represents the solution set of the inequality.

Graph B

Graph B does not accurately represent the solution set of the inequality. The graph shows the solution set as the region between the lines $x=2$ and $x=-4$, but it includes the points $x=2$ and $x=-4$.

Solution Set

The solution set of the inequality $3|x+1| < 9$ is the region between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$. This can be represented graphically as a number line with the lines $x=2$ and $x=-4$ plotted on it. The solution set is the region between these two lines, excluding the points $x=2$ and $x=-4$.

Graphical Representation

The solution set of the inequality $3|x+1| < 9$ can be represented graphically as a number line with the lines $x=2$ and $x=-4$ plotted on it. The solution set is the region between these two lines, excluding the points $x=2$ and $x=-4$.

Conclusion

Based on the graphing instructions, we can conclude that the correct graph for the solution set of the inequality $3|x+1| < 9$ is Graph A.

Final Answer

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Frequently Asked Questions

Q: What is the solution set of the inequality $3|x+1| < 9$?

A: The solution set of the inequality $3|x+1| < 9$ is the region between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$.

Q: How do I graph the solution set of the inequality $3|x+1| < 9$?

A: To graph the solution set, plot the lines $x=2$ and $x=-4$ on a number line. The solution set is the region between these two lines, excluding the points $x=2$ and $x=-4$.

Q: What is the significance of the lines $x=2$ and $x=-4$ in the solution set?

A: The lines $x=2$ and $x=-4$ represent the boundaries of the solution set. The solution set is the region between these two lines, excluding the points $x=2$ and $x=-4$.

Q: How do I determine if a point is in the solution set of the inequality $3|x+1| < 9$?

A: To determine if a point is in the solution set, check if it is between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$.

Q: Can I use a calculator to graph the solution set of the inequality $3|x+1| < 9$?

A: Yes, you can use a calculator to graph the solution set. However, it is recommended to plot the lines $x=2$ and $x=-4$ on a number line to visualize the solution set.

Q: How do I write the solution set of the inequality $3|x+1| < 9$ in interval notation?

A: The solution set of the inequality $3|x+1| < 9$ can be written in interval notation as $(-4, 2)$.

Q: What is the relationship between the solution set of the inequality $3|x+1| < 9$ and the graph of the function $f(x) = 3|x+1|$?

A: The solution set of the inequality $3|x+1| < 9$ is the region between the lines $x=2$ and $x=-4$, excluding the points $x=2$ and $x=-4$. The graph of the function $f(x) = 3|x+1|$ is a V-shaped graph with its vertex at the point $(-1, 0)$.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve other inequalities?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve other inequalities. However, you need to consider the specific inequality and its solution set.

Q: How do I apply the solution set of the inequality $3|x+1| < 9$ to real-world problems?

A: The solution set of the inequality $3|x+1| < 9$ can be applied to real-world problems that involve inequalities and absolute values. For example, you can use the solution set to determine the range of values for a variable in a problem.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve systems of inequalities?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve systems of inequalities. However, you need to consider the specific system of inequalities and its solution set.

Q: How do I extend the solution set of the inequality $3|x+1| < 9$ to other inequalities?

A: You can extend the solution set of the inequality $3|x+1| < 9$ to other inequalities by considering the specific inequality and its solution set. However, you need to be careful when extending the solution set to ensure that it is accurate and complete.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve optimization problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve optimization problems. However, you need to consider the specific optimization problem and its solution set.

Q: How do I apply the solution set of the inequality $3|x+1| < 9$ to linear programming problems?

A: The solution set of the inequality $3|x+1| < 9$ can be applied to linear programming problems that involve inequalities and absolute values. For example, you can use the solution set to determine the range of values for a variable in a problem.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve quadratic programming problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve quadratic programming problems. However, you need to consider the specific quadratic programming problem and its solution set.

Q: How do I extend the solution set of the inequality $3|x+1| < 9$ to other quadratic programming problems?

A: You can extend the solution set of the inequality $3|x+1| < 9$ to other quadratic programming problems by considering the specific quadratic programming problem and its solution set. However, you need to be careful when extending the solution set to ensure that it is accurate and complete.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve nonlinear programming problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve nonlinear programming problems. However, you need to consider the specific nonlinear programming problem and its solution set.

Q: How do I apply the solution set of the inequality $3|x+1| < 9$ to nonlinear programming problems?

A: The solution set of the inequality $3|x+1| < 9$ can be applied to nonlinear programming problems that involve inequalities and absolute values. For example, you can use the solution set to determine the range of values for a variable in a problem.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve integer programming problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve integer programming problems. However, you need to consider the specific integer programming problem and its solution set.

Q: How do I extend the solution set of the inequality $3|x+1| < 9$ to other integer programming problems?

A: You can extend the solution set of the inequality $3|x+1| < 9$ to other integer programming problems by considering the specific integer programming problem and its solution set. However, you need to be careful when extending the solution set to ensure that it is accurate and complete.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve mixed-integer programming problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve mixed-integer programming problems. However, you need to consider the specific mixed-integer programming problem and its solution set.

Q: How do I apply the solution set of the inequality $3|x+1| < 9$ to mixed-integer programming problems?

A: The solution set of the inequality $3|x+1| < 9$ can be applied to mixed-integer programming problems that involve inequalities and absolute values. For example, you can use the solution set to determine the range of values for a variable in a problem.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve stochastic programming problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve stochastic programming problems. However, you need to consider the specific stochastic programming problem and its solution set.

Q: How do I extend the solution set of the inequality $3|x+1| < 9$ to other stochastic programming problems?

A: You can extend the solution set of the inequality $3|x+1| < 9$ to other stochastic programming problems by considering the specific stochastic programming problem and its solution set. However, you need to be careful when extending the solution set to ensure that it is accurate and complete.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve robust optimization problems?

A: Yes, you can use the solution set of the inequality $3|x+1| < 9$ to solve robust optimization problems. However, you need to consider the specific robust optimization problem and its solution set.

Q: How do I apply the solution set of the inequality $3|x+1| < 9$ to robust optimization problems?

A: The solution set of the inequality $3|x+1| < 9$ can be applied to robust optimization problems that involve inequalities and absolute values. For example, you can use the solution set to determine the range of values for a variable in a problem.

Q: Can I use the solution set of the inequality $3|x+1| < 9$ to solve chance-constrained programming problems?

A: Yes, you can use the solution set of