Better understanding of algebra

Oct 13, 2025 by miodeya.com 23 views

Introduction

Inequalities are mathematical expressions that compare two values, often with a greater than or less than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $2(4x - 3) \geq -3(3x) + 5x$ . We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Distribute and Simplify

The first step in solving the inequality is to distribute and simplify the expressions on both sides of the inequality. To do this, we will multiply the numbers outside the parentheses by the terms inside the parentheses.

2(4x - 3) \geq -3(3x) + 5x

Distributing the numbers outside the parentheses, we get:

8x - 6 \geq -9x + 5x

Simplifying the right-hand side of the inequality, we get:

8x - 6 \geq -4x

Step 2: Add 4x to Both Sides

The next step is to add 4x to both sides of the inequality to get all the x terms on one side of the inequality.

8x - 6 + 4x \geq -4x + 4x

Simplifying the left-hand side of the inequality, we get:

12x - 6 \geq 0

Step 3: Add 6 to Both Sides

The next step is to add 6 to both sides of the inequality to get rid of the negative term on the left-hand side.

12x - 6 + 6 \geq 0 + 6

Simplifying the left-hand side of the inequality, we get:

12x \geq 6

Step 4: Divide Both Sides by 12

The final step is to divide both sides of the inequality by 12 to solve for x.

\frac{12x}{12} \geq \frac{6}{12}

Simplifying the left-hand side of the inequality, we get:

x \geq \frac{1}{2}

Conclusion

In conclusion, the solution to the inequality $2(4x - 3) \geq -3(3x) + 5x$ is $x \geq 0.5$ . This means that any value of x greater than or equal to 0.5 will make the inequality true.

Answer

The correct answer is:

A. $x \geq 0.5$

Why is this the correct answer?

This is the correct answer because we solved the inequality by following the steps outlined above. We distributed and simplified the expressions on both sides of the inequality, added 4x to both sides to get all the x terms on one side, added 6 to both sides to get rid of the negative term, and finally divided both sides by 12 to solve for x. The final solution is $x \geq 0.5$ , which means that any value of x greater than or equal to 0.5 will make the inequality true.

What are the other options?

The other options are:

B. $x \geq 2$
C. $(- \infty, 0.5]$
D. $(- \infty, 2]$

These options are incorrect because they do not accurately represent the solution to the inequality. Option B is incorrect because it is a greater value than the correct solution. Option C is incorrect because it is a less than or equal to value, whereas the correct solution is a greater than or equal to value. Option D is incorrect because it is a less than or equal to value, whereas the correct solution is a greater than or equal to value.

Why is this topic important?

This topic is important because it helps students understand how to solve inequalities, which is a fundamental concept in algebra. Solving inequalities is a critical skill that is used in a variety of real-world applications, such as finance, economics, and engineering. By understanding how to solve inequalities, students can apply this knowledge to solve a wide range of problems.

What are the benefits of learning this topic?

The benefits of learning this topic include:

Improved problem-solving skills: By learning how to solve inequalities, students can improve their problem-solving skills and apply this knowledge to a wide range of problems.
Increased confidence: By mastering the concept of solving inequalities, students can increase their confidence in their ability to solve mathematical problems.
Better understanding of algebra: By learning how to solve inequalities, students can gain a deeper understanding of algebra and its applications.

What are the challenges of learning this topic?

The challenges of learning this topic include:

Difficulty with algebraic expressions: Some students may struggle with algebraic expressions, which can make it difficult to solve inequalities.
Confusion with inequality symbols: Some students may confuse the inequality symbols, which can lead to incorrect solutions.
Difficulty with problem-solving: Some students may struggle with problem-solving, which can make it difficult to apply the concept of solving inequalities to real-world problems.

Conclusion

In conclusion, solving inequalities is a critical concept in algebra that helps students understand how to solve mathematical problems. By following the steps outlined above, students can solve inequalities and apply this knowledge to a wide range of problems. The benefits of learning this topic include improved problem-solving skills, increased confidence, and a better understanding of algebra. However, the challenges of learning this topic include difficulty with algebraic expressions, confusion with inequality symbols, and difficulty with problem-solving.