Select The Correct Answer.Which Graph Represents The Solution To The Inequality? − 2.4 ( X − 6 ) ≥ 52.8 -2.4(x-6) \geq 52.8 − 2.4 ( X − 6 ) ≥ 52.8 A. B. C. D.

Understanding the Problem

When solving inequalities, it's essential to understand the concept of graphing solutions. Graphing solutions to inequalities involves representing the solution set on a number line or a coordinate plane. In this article, we'll focus on solving the inequality $-2.4(x-6) \geq 52.8$ and determining which graph represents the solution.

Solving the Inequality

To solve the inequality $-2.4(x-6) \geq 52.8$, we need to follow the order of operations (PEMDAS):

1. Distribute the negative 2.4: $-2.4(x-6) = -2.4x + 14.4$
2. Add 52.8 to both sides: $-2.4x + 14.4 + 52.8 \geq 52.8$
3. Combine like terms: $-2.4x + 67.2 \geq 52.8$
4. Subtract 67.2 from both sides: $-2.4x \geq -14.4$
5. Divide both sides by -2.4: $x \leq 6$

Graphing the Solution

Now that we have the solution to the inequality, we can graph it on a number line. The solution is $x \leq 6$, which means that all values of $x$ less than or equal to 6 are part of the solution set.

Analyzing the Graphs

Let's analyze the graphs provided:

• Graph A: This graph represents the solution set $x \leq 6$. The closed circle at $x=6$ indicates that 6 is included in the solution set.
• Graph B: This graph represents the solution set $x < 6$. The open circle at $x=6$ indicates that 6 is not included in the solution set.
• Graph C: This graph represents the solution set $x \geq 6$. The closed circle at $x=6$ indicates that 6 is included in the solution set.
• Graph D: This graph represents the solution set $x > 6$. The open circle at $x=6$ indicates that 6 is not included in the solution set.

Conclusion

Based on the analysis, the correct graph that represents the solution to the inequality $-2.4(x-6) \geq 52.8$ is Graph A. This graph accurately represents the solution set $x \leq 6$, which includes all values of $x$ less than or equal to 6.

Tips and Tricks

When graphing solutions to inequalities, remember to:

• Use a closed circle to indicate that the endpoint is included in the solution set.
• Use an open circle to indicate that the endpoint is not included in the solution set.
• Label the axis to clearly indicate the variable and the solution set.
• Check your work by plugging in test values to ensure that the solution set is accurate.

Common Mistakes

When solving inequalities, common mistakes include:

• Forgetting to distribute the negative sign or other coefficients.
• Not combining like terms correctly.
• Not checking the direction of the inequality.
• Not graphing the solution set accurately.

Real-World Applications

Solving inequalities has numerous real-world applications, including:

• Finance: Inequality solutions can be used to determine the maximum or minimum values of investments or loans.
• Science: Inequality solutions can be used to model population growth or decay, chemical reactions, or other scientific phenomena.
• Engineering: Inequality solutions can be used to design and optimize systems, such as bridges or buildings.

Conclusion

In conclusion, solving inequalities and graphing solutions is a crucial skill in mathematics. By following the steps outlined in this article, you can accurately solve inequalities and determine which graph represents the solution. Remember to use a closed circle to indicate that the endpoint is included in the solution set, and an open circle to indicate that the endpoint is not included. With practice and patience, you'll become proficient in solving inequalities and graphing solutions.

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 $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I determine the direction of the inequality when solving a linear inequality?

A: To determine the direction of the inequality, you need to consider the sign of the coefficient of the variable. If the coefficient is positive, the inequality is greater than or equal to. If the coefficient is negative, the inequality is less than or equal to.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle on a number line indicates that the endpoint is included in the solution set. An open circle on a number line indicates that the endpoint is not included in the solution set.

Q: How do I graph a solution set on a number line?

A: To graph a solution set on a number line, you need to:

1. Determine the direction of the inequality.
2. Identify the endpoint of the solution set.
3. Use a closed circle to indicate that the endpoint is included in the solution set.
4. Use an open circle to indicate that the endpoint is not included in the solution set.

Q: What is the significance of the axis on a graph?

A: The axis on a graph represents the variable and the solution set. The x-axis represents the variable, and the y-axis represents the solution set.

Q: How do I check my work when solving an inequality?

A: To check your work, you need to:

1. Plug in test values to ensure that the solution set is accurate.
2. Verify the direction of the inequality.
3. Check the endpoint of the solution set.

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

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

1. Forgetting to distribute the negative sign or other coefficients.
2. Not combining like terms correctly.
3. Not checking the direction of the inequality.
4. Not graphing the solution set accurately.

Q: How do I apply inequality solutions to real-world problems?

A: Inequality solutions can be applied to real-world problems in various fields, including finance, science, and engineering. For example, inequality solutions can be used to determine the maximum or minimum values of investments or loans, model population growth or decay, or design and optimize systems.

Q: What are some tips for graphing solutions to inequalities?

A: Some tips for graphing solutions to inequalities include:

1. Use a closed circle to indicate that the endpoint is included in the solution set.
2. Use an open circle to indicate that the endpoint is not included in the solution set.
3. Label the axis to clearly indicate the variable and the solution set.
4. Check your work by plugging in test values to ensure that the solution set is accurate.

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

A: To determine which graph represents the solution to an inequality, you need to:

1. Analyze the graph to determine the direction of the inequality.
2. Identify the endpoint of the solution set.
3. Check the axis to ensure that the variable and solution set are correctly represented.

Conclusion

In conclusion, solving inequalities and graphing solutions is a crucial skill in mathematics. By following the steps outlined in this article, you can accurately solve inequalities and determine which graph represents the solution. Remember to use a closed circle to indicate that the endpoint is included in the solution set, and an open circle to indicate that the endpoint is not included. With practice and patience, you'll become proficient in solving inequalities and graphing solutions.