Which Number Line Shows The Solution To $11x + 14 \ \textless \ -8$?

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Introduction to Inequalities and Number Lines

In mathematics, inequalities are used to compare two or more values. They can be used to represent a wide range of real-world situations, from financial transactions to physical measurements. One of the most common ways to solve inequalities is by using number lines. A number line is a visual representation of the real number system, with numbers arranged in a straight line. By plotting the solution to an inequality on a number line, we can easily identify the values that satisfy the inequality.

Understanding the Given Inequality

The given inequality is $11x + 14 \ \textless \ -8$. To solve this inequality, we need to isolate the variable x. We can start by subtracting 14 from both sides of the inequality, which gives us $11x \ \textless \ -22$. Next, we can divide both sides of the inequality by 11, which gives us $x \ \textless \ -2$. This means that the solution to the inequality is all values of x that are less than -2.

Plotting the Solution on a Number Line

To plot the solution on a number line, we need to identify the values that satisfy the inequality. Since the inequality is of the form $x \ \textless \ -2$, we know that the solution is all values of x that are less than -2. We can plot this on a number line by drawing a line at x = -2 and shading the region to the left of the line. This represents all values of x that are less than -2.

Comparing the Solutions

Now that we have plotted the solution on a number line, we can compare it to the other number lines. We need to find the number line that shows the solution to the given inequality. Let's analyze the other number lines and see which one matches the solution we plotted.

Number Line 1

The first number line shows the solution to the inequality $x \ \textless \ -2$. This is the same solution we plotted earlier, so this number line must show the correct solution.

Number Line 2

The second number line shows the solution to the inequality $x \ \textless \ -3$. This is not the same solution we plotted earlier, so this number line does not show the correct solution.

Number Line 3

The third number line shows the solution to the inequality $x \ \textless \ -1$. This is not the same solution we plotted earlier, so this number line does not show the correct solution.

Conclusion

Based on our analysis, we can conclude that the number line that shows the solution to the given inequality is the first number line. This number line correctly represents the solution to the inequality $11x + 14 \ \textless \ -8$.

Final Answer

The final answer is: Number Line 1

Step-by-Step Solution

  1. Subtract 14 from both sides of the inequality: $11x + 14 \ \textless \ -8$
  2. Subtract 14 from both sides of the inequality: $11x \ \textless \ -22$
  3. Divide both sides of the inequality by 11: $x \ \textless \ -2$
  4. Plot the solution on a number line: Draw a line at x = -2 and shade the region to the left of the line.
  5. Compare the solutions: Analyze the other number lines and see which one matches the solution we plotted.

Frequently Asked Questions

  • Q: What is the solution to the inequality $11x + 14 \ \textless \ -8$? A: The solution to the inequality is all values of x that are less than -2.
  • Q: How do I plot the solution on a number line? A: To plot the solution on a number line, draw a line at x = -2 and shade the region to the left of the line.
  • Q: Which number line shows the correct solution? A: The first number line shows the correct solution.

Related Topics

  • Inequalities
  • Number lines
  • Algebra
  • Mathematics

References

Q: What is an inequality?

A: An inequality is a statement that compares two or more values using a mathematical symbol, such as <, >, ≤, or ≥.

Q: What is a number line?

A: A number line is a visual representation of the real number system, with numbers arranged in a straight line.

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. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: How do I plot the solution to an inequality on a number line?

A: To plot the solution to an inequality on a number line, you need to identify the values that satisfy the inequality. You can do this by drawing a line at the value that makes the inequality true, and shading the region to the left or right of the line, depending on the direction of the inequality symbol.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality, such as < or >, means that the value on one side of the inequality symbol is strictly less than or strictly greater than the value on the other side. A non-strict inequality, such as ≤ or ≥, means that the value on one side of the inequality symbol is less than or equal to or greater than or equal to the value on the other side.

Q: How do I compare two or more inequalities?

A: To compare two or more inequalities, you need to determine which values satisfy each inequality. You can do this by plotting the solutions to each inequality on a number line, and then comparing the regions that are shaded.

Q: What is the solution to the inequality $11x + 14 \ \textless \ -8$?

A: The solution to the inequality is all values of x that are less than -2.

Q: How do I determine the solution to an inequality with multiple variables?

A: To determine the solution to an inequality with multiple variables, you need to isolate each variable on one side of the inequality symbol. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

A: A linear inequality, such as $x \ \textless \ 2$, is an inequality that can be written in the form $ax + b \ \textless \ c$, where a, b, and c are constants. A nonlinear inequality, such as $x^2 \ \textless \ 4$, is an inequality that cannot be written in the form $ax + b \ \textless \ c$.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the values that satisfy all of the inequalities in the system. You can do this by plotting the solutions to each inequality on a number line, and then identifying the region that is common to all of the inequalities.

Q: What is the importance of inequalities in real-world applications?

A: Inequalities are used to model a wide range of real-world situations, from financial transactions to physical measurements. They are used to compare values, determine relationships between variables, and make predictions about future outcomes.

Q: How do I use inequalities to make predictions about future outcomes?

A: To use inequalities to make predictions about future outcomes, you need to identify the values that satisfy the inequality, and then use those values to make predictions about future outcomes. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to make predictions about future outcomes.

Q: What are some common applications of inequalities in real-world situations?

A: Some common applications of inequalities in real-world situations include:

  • Financial transactions: Inequalities are used to compare the costs of different financial transactions, such as loans and credit cards.
  • Physical measurements: Inequalities are used to compare the measurements of different physical objects, such as length and width.
  • Science: Inequalities are used to compare the values of different scientific variables, such as temperature and pressure.
  • Engineering: Inequalities are used to compare the values of different engineering variables, such as stress and strain.

Q: How do I use inequalities to compare the values of different variables?

A: To use inequalities to compare the values of different variables, you need to identify the values that satisfy the inequality, and then use those values to compare the variables. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to compare the variables.

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

A: Some common mistakes to avoid when working with inequalities include:

  • Not isolating the variable on one side of the inequality symbol.
  • Not plotting the solution to the inequality on a number line.
  • Not comparing the values of different variables using inequalities.
  • Not using inequalities to make predictions about future outcomes.

Q: How do I use inequalities to make predictions about future outcomes in a real-world situation?

A: To use inequalities to make predictions about future outcomes in a real-world situation, you need to identify the values that satisfy the inequality, and then use those values to make predictions about future outcomes. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to make predictions about future outcomes.

Q: What are some common applications of inequalities in real-world situations that involve multiple variables?

A: Some common applications of inequalities in real-world situations that involve multiple variables include:

  • Financial transactions: Inequalities are used to compare the costs of different financial transactions, such as loans and credit cards.
  • Physical measurements: Inequalities are used to compare the measurements of different physical objects, such as length and width.
  • Science: Inequalities are used to compare the values of different scientific variables, such as temperature and pressure.
  • Engineering: Inequalities are used to compare the values of different engineering variables, such as stress and strain.

Q: How do I use inequalities to compare the values of different variables in a real-world situation that involves multiple variables?

A: To use inequalities to compare the values of different variables in a real-world situation that involves multiple variables, you need to identify the values that satisfy the inequality, and then use those values to compare the variables. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to compare the variables.

Q: What are some common mistakes to avoid when working with inequalities in a real-world situation that involves multiple variables?

A: Some common mistakes to avoid when working with inequalities in a real-world situation that involves multiple variables include:

  • Not isolating the variables on one side of the inequality symbol.
  • Not plotting the solutions to the inequality on a number line.
  • Not comparing the values of different variables using inequalities.
  • Not using inequalities to make predictions about future outcomes.

Q: How do I use inequalities to make predictions about future outcomes in a real-world situation that involves multiple variables?

A: To use inequalities to make predictions about future outcomes in a real-world situation that involves multiple variables, you need to identify the values that satisfy the inequality, and then use those values to make predictions about future outcomes. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to make predictions about future outcomes.

Q: What are some common applications of inequalities in real-world situations that involve nonlinear inequalities?

A: Some common applications of inequalities in real-world situations that involve nonlinear inequalities include:

  • Financial transactions: Inequalities are used to compare the costs of different financial transactions, such as loans and credit cards.
  • Physical measurements: Inequalities are used to compare the measurements of different physical objects, such as length and width.
  • Science: Inequalities are used to compare the values of different scientific variables, such as temperature and pressure.
  • Engineering: Inequalities are used to compare the values of different engineering variables, such as stress and strain.

Q: How do I use inequalities to compare the values of different variables in a real-world situation that involves nonlinear inequalities?

A: To use inequalities to compare the values of different variables in a real-world situation that involves nonlinear inequalities, you need to identify the values that satisfy the inequality, and then use those values to compare the variables. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to compare the variables.

Q: What are some common mistakes to avoid when working with inequalities in a real-world situation that involves nonlinear inequalities?

A: Some common mistakes to avoid when working with inequalities in a real-world situation that involves nonlinear inequalities include:

  • Not isolating the variables on one side of the inequality symbol.
  • Not plotting the solutions to the inequality on a number line.
  • Not comparing the values of different variables using inequalities.
  • Not using inequalities to make predictions about future outcomes.

Q: How do I use inequalities to make predictions about future outcomes in a real-world situation that involves nonlinear inequalities?

A: To use inequalities to make predictions about future outcomes in a real-world situation that involves nonlinear inequalities, you need to identify the values that satisfy the inequality, and then use those values to make predictions about future outcomes. You can do this by plotting the solutions to the inequality on a number line, and then using the region that is shaded to make predictions about future outcomes.

Q: What are some common applications of inequalities in real-world situations that involve systems of inequalities?

A: Some common applications of inequalities in real