Which Inequality Shows That Twelve Times The Sum Of Three Plus $c$ Is Greater Than 27?A. 12 ( 3 + C ) \textgreater 27 12(3+c) \ \textgreater \ 27 12 ( 3 + C ) \textgreater 27 B. ( 12 × 3 ) + C \textgreater 27 (12 \times 3) + C \ \textgreater \ 27 ( 12 × 3 ) + C \textgreater 27 C. ( 12 + C ) × 3 \textgreater 27 (12+c) \times 3 \ \textgreater \ 27 ( 12 + C ) × 3 \textgreater 27 D.
Understanding Inequalities
In mathematics, inequalities are used to compare two or more values. They are often used to represent real-world situations where we need to determine if one value is greater than, less than, or equal to another value. In this article, we will focus on solving inequalities, specifically the inequality that shows that twelve times the sum of three plus $c$ is greater than 27.
The Basics of Inequalities
An inequality is a statement that compares two values using a mathematical symbol, such as >, <, ≥, or ≤. For example, the statement $x > 5$ means that $x$ is greater than 5. Inequalities can be used to represent a wide range of real-world situations, such as comparing the cost of two items, determining the maximum or minimum value of a quantity, or finding the range of values that satisfy a given condition.
Solving Inequalities
To solve an inequality, we need to isolate the variable on one side of the inequality symbol. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value. For example, to solve the inequality $x + 2 > 5$, we can subtract 2 from both sides to get $x > 3$.
The Inequality in Question
The inequality in question is $12(3+c) \ \textgreater \ 27$. This inequality states that twelve times the sum of three plus $c$ is greater than 27. To solve this inequality, we need to isolate the variable $c$ on one side of the inequality symbol.
Step 1: Distribute the 12
The first step in solving the inequality is to distribute the 12 to the terms inside the parentheses. This gives us $36 + 12c \ \textgreater \ 27$.
Step 2: Subtract 36 from Both Sides
Next, we subtract 36 from both sides of the inequality to get $12c \ \textgreater \ -9$.
Step 3: Divide Both Sides by 12
Finally, we divide both sides of the inequality by 12 to get $c \ \textgreater \ -\frac{3}{4}$.
Conclusion
In conclusion, the inequality that shows that twelve times the sum of three plus $c$ is greater than 27 is $12(3+c) \ \textgreater \ 27$. By following the steps outlined above, we can solve this inequality and determine the value of $c$ that satisfies the given condition.
Comparison of Options
Let's compare the options given in the problem to see which one is correct.
- Option A:
- Option B:
- Option C:
- Option D: (not given)
Option A is the correct answer because it accurately represents the inequality that shows that twelve times the sum of three plus $c$ is greater than 27.
Real-World Applications
Inequalities have many real-world applications, such as:
- Comparing the cost of two items
- Determining the maximum or minimum value of a quantity
- Finding the range of values that satisfy a given condition
- Solving optimization problems
Inequalities are an essential tool in mathematics and are used to represent a wide range of real-world situations.
Tips and Tricks
Here are some tips and tricks for solving inequalities:
- Always read the inequality carefully and understand what it is saying.
- Use inverse operations to isolate the variable on one side of the inequality symbol.
- Be careful when multiplying or dividing both sides of the inequality by a negative value.
- Use a calculator to check your answers and make sure they are correct.
By following these tips and tricks, you can become proficient in solving inequalities and apply them to real-world situations.
Conclusion
In conclusion, solving inequalities is an essential skill in mathematics that has many real-world applications. By following the steps outlined above, you can solve inequalities and determine the value of the variable that satisfies the given condition. Remember to always read the inequality carefully, use inverse operations to isolate the variable, and be careful when multiplying or dividing both sides of the inequality by a negative value. With practice and patience, you can become proficient in solving inequalities and apply them to real-world situations.
Q: What is an inequality?
A: An inequality is a statement that compares two values using a mathematical symbol, such as >, <, ≥, or ≤. For example, the statement $x > 5$ means that $x$ is greater than 5.
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate the variable on one side of the inequality symbol. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Q: What is the difference between an inequality and an equation?
A: An equation is a statement that says two values are equal, while an inequality is a statement that compares two values using a mathematical symbol. For example, the equation $x + 2 = 5$ means that $x + 2$ is equal to 5, while the inequality $x + 2 > 5$ means that $x + 2$ is greater than 5.
Q: How do I determine the direction of the 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 symbol will point to the right (e.g., $x > 5$). If the coefficient is negative, the inequality symbol will point to the left (e.g., $x < 5$).
Q: Can I multiply or divide both sides of an inequality by a negative value?
A: No, you cannot multiply or divide both sides of an inequality by a negative value. This will change the direction of the inequality, which can lead to incorrect solutions.
Q: How do I solve a compound inequality?
A: A compound inequality is an inequality that contains two or more inequality symbols. To solve a compound inequality, you need to solve each inequality separately and then combine the solutions.
Q: Can I use a calculator to solve an inequality?
A: Yes, you can use a calculator to solve an inequality. However, you need to make sure that the calculator is set to the correct mode (e.g., fraction mode) and that you are using the correct operations (e.g., division instead of multiplication).
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. If the inequality is of the form $x > a$, you will plot a point to the right of $a$. If the inequality is of the form $x < a$, you will plot a point to the left of $a$.
Q: Can I use a graphing calculator to graph an inequality?
A: Yes, you can use a graphing calculator to graph an inequality. However, you need to make sure that the calculator is set to the correct mode (e.g., function mode) and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the solution to an inequality?
A: To determine the solution to an inequality, you need to isolate the variable on one side of the inequality symbol and then determine the values of the variable that satisfy the inequality.
Q: Can I use a table to solve an inequality?
A: Yes, you can use a table to solve an inequality. However, you need to make sure that the table is set up correctly and that you are using the correct operations (e.g., adding or subtracting the same value to both sides of the inequality).
Q: How do I check my solution to an inequality?
A: To check your solution to an inequality, you need to plug the solution back into the original inequality and make sure that it is true. If the solution is not true, you need to re-solve the inequality and try again.
Q: Can I use a computer program to solve an inequality?
A: Yes, you can use a computer program to solve an inequality. However, you need to make sure that the program is set up correctly and that you are using the correct operations (e.g., using a linear programming solver).
Q: How do I determine the domain of an inequality?
A: To determine the domain of an inequality, you need to consider the values of the variable that satisfy the inequality. The domain of an inequality is the set of all values of the variable that make the inequality true.
Q: Can I use a graph to determine the domain of an inequality?
A: Yes, you can use a graph to determine the domain of an inequality. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the range of an inequality?
A: To determine the range of an inequality, you need to consider the values of the variable that satisfy the inequality. The range of an inequality is the set of all values of the variable that make the inequality true.
Q: Can I use a graph to determine the range of an inequality?
A: Yes, you can use a graph to determine the range of an inequality. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the solution to a system of inequalities?
A: To determine the solution to a system of inequalities, you need to solve each inequality separately and then combine the solutions. The solution to a system of inequalities is the set of all values of the variables that make all of the inequalities true.
Q: Can I use a graph to determine the solution to a system of inequalities?
A: Yes, you can use a graph to determine the solution to a system of inequalities. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequalities instead of the equations).
Q: How do I determine the solution to a linear programming problem?
A: To determine the solution to a linear programming problem, you need to solve the system of inequalities that represents the problem. The solution to a linear programming problem is the set of all values of the variables that make all of the inequalities true and maximize or minimize the objective function.
Q: Can I use a computer program to determine the solution to a linear programming problem?
A: Yes, you can use a computer program to determine the solution to a linear programming problem. However, you need to make sure that the program is set up correctly and that you are using the correct operations (e.g., using a linear programming solver).
Q: How do I determine the solution to a quadratic inequality?
A: To determine the solution to a quadratic inequality, you need to solve the quadratic equation that represents the inequality. The solution to a quadratic inequality is the set of all values of the variable that make the inequality true.
Q: Can I use a graph to determine the solution to a quadratic inequality?
A: Yes, you can use a graph to determine the solution to a quadratic inequality. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the solution to a rational inequality?
A: To determine the solution to a rational inequality, you need to solve the rational equation that represents the inequality. The solution to a rational inequality is the set of all values of the variable that make the inequality true.
Q: Can I use a graph to determine the solution to a rational inequality?
A: Yes, you can use a graph to determine the solution to a rational inequality. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the solution to a polynomial inequality?
A: To determine the solution to a polynomial inequality, you need to solve the polynomial equation that represents the inequality. The solution to a polynomial inequality is the set of all values of the variable that make the inequality true.
Q: Can I use a graph to determine the solution to a polynomial inequality?
A: Yes, you can use a graph to determine the solution to a polynomial inequality. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the solution to a trigonometric inequality?
A: To determine the solution to a trigonometric inequality, you need to solve the trigonometric equation that represents the inequality. The solution to a trigonometric inequality is the set of all values of the variable that make the inequality true.
Q: Can I use a graph to determine the solution to a trigonometric inequality?
A: Yes, you can use a graph to determine the solution to a trigonometric inequality. However, you need to make sure that the graph is set up correctly and that you are using the correct operations (e.g., plotting the inequality instead of the equation).
Q: How do I determine the solution to a logarithmic inequality?
A: To determine the solution to a logarithmic inequality, you need to solve the logarithmic equation that represents the inequality. The solution to a logarithmic inequality is the set of all values of the variable that make the inequality true.