Finding the Intersection of the Two Inequalities

Oct 13, 2025 by windailys.com 11 views

Introduction

Inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will delve into the world of inequalities and explore how to solve them. We will focus on a specific type of inequality, where we have two inequalities with the same variable, and we need to find the range of values that satisfies both inequalities.

Understanding the Problem

The problem we are given is:

6x < 2 - 3x

and

x - 7 < 3x

We need to find the range of values of $x$ that satisfies both inequalities.

Step 1: Solving the First Inequality

To solve the first inequality, we need to isolate the variable $x$ . We can start by adding $3x$ to both sides of the inequality:

6x + 3x < 2 - 3x + 3x

This simplifies to:

9x < 2

Next, we can divide both sides of the inequality by $9$ :

x < \frac{2}{9}

So, the solution to the first inequality is $x < \frac{2}{9}$ .

Step 2: Solving the Second Inequality

To solve the second inequality, we need to isolate the variable $x$ . We can start by adding $7$ to both sides of the inequality:

x + 7 < 3x + 7

This simplifies to:

7 < 2x + 7

Next, we can subtract $7$ from both sides of the inequality:

0 < 2x

Finally, we can divide both sides of the inequality by $2$ :

0 < x

So, the solution to the second inequality is $x > 0$ .

Finding the Intersection of the Two Inequalities

Now that we have solved both inequalities, we need to find the range of values that satisfies both inequalities. To do this, we need to find the intersection of the two inequalities.

The first inequality is $x < \frac{2}{9}$ , and the second inequality is $x > 0$ . To find the intersection, we need to find the values of $x$ that satisfy both inequalities.

Since $x > 0$ is a more restrictive inequality than $x < \frac{2}{9}$ , we can conclude that the intersection of the two inequalities is $0 < x < \frac{2}{9}$ .

Conclusion

In this article, we have solved two inequalities and found the range of values that satisfies both inequalities. We have shown that the intersection of the two inequalities is $0 < x < \frac{2}{9}$ . This is the range of values that satisfies both inequalities.

Final Answer

The final answer is $\boxed{0 < x < \frac{2}{9}}$ .

Additional Tips and Tricks

When solving inequalities, it's essential to isolate the variable on one side of the inequality.
When adding or subtracting the same value to both sides of an inequality, the direction of the inequality remains the same.
When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality is reversed.
When solving inequalities, it's essential to consider the direction of the inequality and the restrictions on the variable.

Common Mistakes to Avoid

Not isolating the variable on one side of the inequality.
Not considering the direction of the inequality.
Not checking the restrictions on the variable.
Not using the correct operations to solve the inequality.

Real-World Applications

Inequalities have numerous real-world applications, including:

Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Inequalities are used to model economic systems, including supply and demand, inflation, and unemployment.

Conclusion

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two expressions using a relation such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by a positive value.

Q: What is the difference between an inequality and an equation?

A: An equation is a mathematical statement that states that two expressions are equal. An inequality, on the other hand, states that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

Q: How do I know which direction to go when solving an inequality?

A: When solving an inequality, you need to consider the direction of the inequality. If the inequality is greater than (>), you need to go in the positive direction. If the inequality is less than (<), you need to go in the negative direction.

Q: What is the intersection of two inequalities?

A: The intersection of two inequalities is the range of values that satisfies both inequalities. To find the intersection, you need to find the values of the variable that satisfy both inequalities.

Q: How do I find the intersection of two inequalities?

A: To find the intersection of two inequalities, you need to solve both inequalities separately and then find the values of the variable that satisfy both inequalities. You can do this by finding the common values of the variable that satisfy both inequalities.

Q: What is the range of values that satisfies both inequalities?

A: The range of values that satisfies both inequalities is the intersection of the two inequalities. This is the range of values that satisfies both inequalities.

Q: How do I check my answer?

A: To check your answer, you need to plug in the values of the variable into both inequalities and make sure that the inequality is true. If the inequality is true, then your answer is correct.

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

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

Not isolating the variable on one side of the inequality.
Not considering the direction of the inequality.
Not checking the restrictions on the variable.
Not using the correct operations to solve the inequality.

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

A: Inequalities have numerous real-world applications, including:

Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Inequalities are used to model economic systems, including supply and demand, inflation, and unemployment.

Q: What are some tips and tricks for solving inequalities?

A: Some tips and tricks for solving inequalities include:

Isolating the variable on one side of the inequality.
Considering the direction of the inequality.
Checking the restrictions on the variable.
Using the correct operations to solve the inequality.

Conclusion

In conclusion, solving inequalities is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve inequalities and find the range of values that satisfies both inequalities. Remember to isolate the variable, consider the direction of the inequality, and check the restrictions on the variable. With practice and patience, you can become proficient in solving inequalities and apply them to real-world problems.

Final Answer

The final answer is $\boxed{0 < x < \frac{2}{9}}$ .

Additional Resources

Khan Academy: Inequalities
Mathway: Inequalities
Wolfram Alpha: Inequalities

Common Inequalities

$x > a$
$x < a$
$x \geq a$
$x \leq a$
$x + a > b$
$x + a < b$
$x + a \geq b$
$x + a \leq b$

Real-World Applications

Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Inequalities are used to model economic systems, including supply and demand, inflation, and unemployment.