Solve Each Inequality. Then Compare The Solutions.1. $5x + 11 \ \textless \ 21$2. − 2 X + 22 \textless 18 -2x + 22 \ \textless \ 18 − 2 X + 22 \textless 18

In mathematics, inequalities are used to compare the values of two or more expressions. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will solve two inequalities and compare their solutions.

Inequality 1: $5x + 11 \ \textless \ 21$

To solve the inequality $5x + 11 \ \textless \ 21$, we need to isolate the variable $x$. We can start by subtracting 11 from both sides of the inequality.

5x + 11 - 11 \  \textless \  21 - 11

This simplifies to:

5x \  \textless \  10

Next, we can divide both sides of the inequality by 5 to solve for $x$.

\frac{5x}{5} \  \textless \  \frac{10}{5}

This simplifies to:

x \  \textless \  2

Therefore, the solution to the inequality $5x + 11 \ \textless \ 21$ is $x \ \textless \ 2$.

Inequality 2: $-2x + 22 \ \textless \ 18$

To solve the inequality $-2x + 22 \ \textless \ 18$, we need to isolate the variable $x$. We can start by subtracting 22 from both sides of the inequality.

-2x + 22 - 22 \  \textless \  18 - 22

This simplifies to:

-2x \  \textless \  -4

Next, we can divide both sides of the inequality by -2 to solve for $x$. When we divide by a negative number, we need to reverse the direction of the inequality.

\frac{-2x}{-2} \  \textgreater \  \frac{-4}{-2}

This simplifies to:

x \  \textgreater \  2

Therefore, the solution to the inequality $-2x + 22 \ \textless \ 18$ is $x \ \textgreater \ 2$.

Comparing the Solutions

Now that we have solved both inequalities, we can compare their solutions. The solution to the first inequality is $x \ \textless \ 2$, while the solution to the second inequality is $x \ \textgreater \ 2$. These two solutions are mutually exclusive, meaning that they cannot both be true at the same time.

In other words, if $x$ is less than 2, it cannot be greater than 2, and vice versa. Therefore, the two solutions do not overlap, and there is no value of $x$ that satisfies both inequalities.

Conclusion

In this article, we solved two inequalities and compared their solutions. We found that the solutions to the two inequalities are mutually exclusive, meaning that they cannot both be true at the same time. This highlights the importance of carefully solving and comparing inequalities in mathematics.

Key Takeaways

• To solve an inequality, we need to isolate the variable.
• When dividing by a negative number, we need to reverse the direction of the inequality.
• The solutions to two inequalities can be mutually exclusive, meaning that they cannot both be true at the same time.

Practice Problems

1. Solve the inequality $3x - 5 \ \textless \ 11$.
2. Solve the inequality $x + 2 \ \textless \ 7$.
3. Solve the inequality $-4x + 15 \ \textless \ 3$.

Answer Key

1. $x \ \textless \ 4$
2. $x \ \textless \ 5$
3. $x \ \textless \ -3$

Additional Resources

For more practice problems and additional resources, check out the following websites:

• Khan Academy: Inequalities
• Mathway: Inequalities
• IXL: Inequalities

In this article, we will answer some of the most frequently asked questions about solving and comparing inequalities.

Q: What is an inequality?

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

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing the same operations on both sides of the inequality. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality.

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 < c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c < d, where a, b, c, and d are constants.

Q: How do I compare the solutions to two inequalities?

A: To compare the solutions to two inequalities, you need to determine if the solutions overlap or if they are mutually exclusive. If the solutions overlap, then there are values of the variable that satisfy both inequalities. If the solutions are mutually exclusive, then there are no values of the variable that satisfy both inequalities.

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

A: A strict inequality is an inequality that uses a strict comparison operator such as < or >. A non-strict inequality is an inequality that uses a non-strict comparison operator such as ≤ or ≥.

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 strict, you need to plot an open circle at the point. If the inequality is non-strict, you need to plot a closed circle at the point.

Q: What is the importance of solving and comparing inequalities?

A: Solving and comparing inequalities is an important skill in mathematics because it allows you to determine the values of a variable that satisfy a given inequality. This is useful in a variety of real-world applications, such as finance, science, and engineering.

Q: How can I practice solving and comparing inequalities?

A: You can practice solving and comparing inequalities by working through practice problems and exercises. You can also use online resources such as Khan Academy, Mathway, and IXL to practice solving and comparing inequalities.

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

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

• Not isolating the variable
• Not reversing the direction of the inequality when dividing by a negative number
• Not considering the direction of the inequality when comparing solutions
• Not graphing the inequality on a number line

Conclusion

In this article, we have answered some of the most frequently asked questions about solving and comparing inequalities. We have covered topics such as the definition of an inequality, how to solve an inequality, and how to compare the solutions to two inequalities. We have also discussed the importance of solving and comparing inequalities and provided some tips for practicing and avoiding common mistakes.

Key Takeaways

• An inequality is a mathematical statement that compares two or more expressions using a comparison operator.
• To solve an inequality, you need to isolate the variable by performing the same operations on both sides of the inequality.
• To compare the solutions to two inequalities, you need to determine if the solutions overlap or if they are mutually exclusive.
• Solving and comparing inequalities is an important skill in mathematics because it allows you to determine the values of a variable that satisfy a given inequality.

Practice Problems

1. Solve the inequality $2x + 3 \ \textless \ 5$.
2. Solve the inequality $x - 2 \ \textless \ 3$.
3. Solve the inequality $-3x + 2 \ \textless \ 1$.

Answer Key

1. $x \ \textless \ 1$
2. $x \ \textless \ 5$
3. $x \ \textless \ -\frac{1}{3}$

Additional Resources

For more practice problems and additional resources, check out the following websites:

• Khan Academy: Inequalities
• Mathway: Inequalities
• IXL: Inequalities

By following these resources and practicing regularly, you can improve your skills in solving and comparing inequalities.