When it comes to solving inequalities, understanding the underlying principles and techniques is essential. In this article, we will delve into the specifics of solving the inequality 2(4x – 3) ≥ –3(3x) + 5x. Additionally, we will explore the range of values that x must satisfy based on the given conditions X ≥ 0.5 and X ≥ 2.
With a relaxed writing style and a focus on clarity, this article aims to provide a comprehensive guide on tackling this inequality problem. Whether you’re a student grappling with algebraic concepts or an individual seeking to refresh your mathematical skills, this article will equip you with the knowledge to solve the inequality effectively.
Throughout this article, we will break down the steps and explain the reasoning behind each one. By offering a clear and concise approach, we aim to make the process of solving the inequality 2(4x – 3) ≥ –3(3x) + 5x accessible to all readers. So let’s dive in and discover how to find the solution for x, ensuring our values fall within the range defined by X ≥ 0.5 and X ≥ 2.
Solving the Inequality: 2(4x – 3) ≥ –3(3x) + 5x
Introduction
Inequality equations are mathematical expressions that establish a relationship between two values that are not equal. Solving an inequality involves finding all the values of the variable that make the inequality true. In this article, we will explore the steps involved in solving the inequality 2(4x – 3) ≥ –3(3x) + 5x, given the conditions X ≥ 0.5 and X ≥ 2.
Step 1: Distribute the terms
The first step in solving the given inequality is to distribute the terms on both sides of the equation. This will help simplify the expression and make it easier to manipulate. Distributing the terms, we get: 8x – 6 ≥ -9x + 5x.
Step 2: Combine like terms
Once the terms have been distributed, the next step is to combine like terms on both sides of the equation. In this case, we have 8x and -9x + 5x. Combining these terms, we get: 8x – 6 ≥ -4x.
Step 3: Move all terms to one side
To further simplify the equation, we need to move all the terms involving x to one side of the equation. In this case, we want to isolate the variable x to determine the valid values. Moving the terms involving x to one side, we get: 8x + 4x ≥ 6.
Step 4: Combine like terms
Next, we combine the like terms on the left side of the equation. Adding 8x and 4x, we get: 12x ≥ 6.
Step 5: Divide by the coefficient of x
To solve for x, we divide both sides of the equation by the coefficient of x, which in this case is 12. Dividing 12x by 12, we get: x ≥ 0.5.
Step 6: Check the conditions
After obtaining the solution x ≥ 0.5, we need to check if it satisfies the given conditions X ≥ 0.5 and X ≥ 2. Both conditions are met since 0.5 is greater than or equal to 0.5 and also greater than or equal to 2.
Step 7: Final Solution
Therefore, the solution to the inequality 2(4x – 3) ≥ –3(3x) + 5x, given the conditions X ≥ 0.5 and X ≥ 2, is x ≥ 0.5. The solution represents all the possible values of x that satisfy the inequality and meet the given conditions.
Summary
Solving an inequality involves several steps, including distributing the terms, combining like terms, moving all terms to one side, dividing by the coefficient of the variable, and checking the given conditions. By following these steps, we are able to determine the valid values of x that satisfy the inequality. In this case, the solution is x ≥ 0.5, which fulfills the given conditions X ≥ 0.5 and X ≥ 2.
Subtitle: “Solving the Inequality: 2(4x – 3) ≥ –3(3x) + 5x? X ≥ 0.5 X ≥ 2 (–∞, 0.5] (–∞, 2]”
Keyword: Solving the Inequality
Steps to solve the inequality:
1. Distribute the multiplication on both sides of the inequality to remove the parentheses.
2(4x – 3) ≥ -3(3x) + 5x simplifies to 8x – 6 ≥ -9x + 5x.
2. Combine like terms on both sides of the inequality.
On the left side, we have 8x – 6, and on the right side, -9x + 5x simplifies to -4x.
The inequality becomes 8x – 6 ≥ -4x.
3. Add 4x to both sides of the inequality to isolate the variable on one side.
8x – 6 + 4x ≥ -4x + 4x simplifies to 12x – 6 ≥ 0.
4. Add 6 to both sides of the inequality.
12x – 6 + 6 ≥ 0 + 6 simplifies to 12x ≥ 6.
5. Divide both sides of the inequality by 12 to solve for x.
12x/12 ≥ 6/12 simplifies to x ≥ 0.5.
6. Check the given conditions.
The inequality states x ≥ 0.5. Since the initial restriction is x ≥ 2, we need to verify if x ≥ 0.5 satisfies this condition as well.
Since 0.5 is greater than or equal to 2, the condition x ≥ 0.5 satisfies the initial restriction.
7. Determine the solution set.
The solution set to the inequality 2(4x – 3) ≥ –3(3x) + 5x is x ≥ 0.5.
8. Expressing the solution set in interval notation.
The interval notation for x ≥ 0.5 is (0.5, ∞).
What is the inequality being solved in this question?
The inequality being solved is 2(4x – 3) ≥ –3(3x) + 5x.
What are the restrictions for the variable x in this inequality?
The restrictions for the variable x in this inequality are x ≥ 0.5 and x ≥ 2.
What is the solution set for the inequality?
The solution set for the inequality is (–∞, 2].
What does the notation (–∞, 0.5] mean?
The notation (–∞, 0.5] represents all real numbers less than or equal to 0.5, including 0.5.
What does the notation (–∞, 2] mean?
The notation (–∞, 2] represents all real numbers less than or equal to 2, including 2.
Conclusion
The inequality 2(4x – 3) ≥ –3(3x) + 5x has a solution set of x values that are greater than or equal to 2. The notation (–∞, 2] represents this solution set.