Solve Two-thirds X Minus One-fifth Greater-than 1.

Solving the Inequality: (2/3)x - (1/5) > 1

This article will guide you through solving the inequality (2/3)x - (1/5) > 1, step-by-step. We'll cover the fundamental algebraic principles involved, explore different methods for solving the inequality, and discuss the interpretation of the solution set. Furthermore, we'll delve into related concepts to provide a comprehensive understanding of solving linear inequalities.

Understanding Linear Inequalities

Before we tackle the specific inequality, let's review the basics of linear inequalities. A linear inequality is an inequality that involves a linear expression, meaning an expression where the variable's highest power is 1. These inequalities are represented using symbols like:

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

Unlike equations, which have a single solution, linear inequalities typically have an infinite number of solutions, forming a solution set. This solution set represents all the values of the variable that satisfy the inequality.

Solving the Inequality: (2/3)x - (1/5) > 1

Now, let's solve the inequality (2/3)x - (1/5) > 1. Our goal is to isolate the variable 'x' on one side of the inequality sign. We'll achieve this by applying algebraic operations to both sides of the inequality, ensuring we maintain the inequality's direction.

Step 1: Eliminate Fractions

The presence of fractions can make the calculations more complex. To simplify the inequality, let's eliminate the fractions by finding the least common multiple (LCM) of the denominators 3 and 5. The LCM of 3 and 5 is 15. We multiply both sides of the inequality by 15:

15 * [(2/3)x - (1/5)] > 15 * 1

This simplifies to:

10x - 3 > 15

Step 2: Isolate the Variable Term

Next, we need to isolate the term containing 'x'. We can do this by adding 3 to both sides of the inequality:

10x - 3 + 3 > 15 + 3

This simplifies to:

10x > 18

Step 3: Solve for x

Finally, to solve for 'x', we divide both sides of the inequality by 10:

10x / 10 > 18 / 10

This gives us:

x > 1.8 or x > 9/5

Therefore, the solution to the inequality (2/3)x - (1/5) > 1 is x > 1.8 (or x > 9/5). This means any value of 'x' greater than 1.8 will satisfy the original inequality.

Representing the Solution Set

The solution set can be represented in several ways:

• Interval Notation: (1.8, ∞) This notation indicates that the solution set includes all values from 1.8 (exclusive) to infinity.
• Set-Builder Notation: {x | x > 1.8} This notation reads as "the set of all x such that x is greater than 1.8".
• Number Line: A number line can visually represent the solution set. We would draw an open circle at 1.8 and shade the region to the right, indicating all values greater than 1.8.

Verifying the Solution

It's always a good practice to verify the solution. Let's test a value greater than 1.8, say x = 2:

(2/3)(2) - (1/5) = 4/3 - 1/5 = (20 - 3)/15 = 17/15 ≈ 1.13

Since 1.13 > 1, our solution is correct. Now let's test a value less than 1.8, say x = 1:

(2/3)(1) - (1/5) = 2/3 - 1/5 = (10 - 3)/15 = 7/15 ≈ 0.47

Since 0.47 is not greater than 1, this confirms that our solution is accurate.

Related Concepts and Extensions

Understanding linear inequalities lays the groundwork for more advanced mathematical concepts. Here are some related topics to explore:

Systems of Linear Inequalities:

This involves solving multiple linear inequalities simultaneously. The solution set is the region where all inequalities are satisfied. Graphing is often used to visualize the solution region.

Linear Programming:

Linear programming uses linear inequalities to optimize an objective function subject to constraints. This is widely applied in operations research, economics, and other fields.

Nonlinear Inequalities:

These inequalities involve variables raised to powers other than 1. Solving nonlinear inequalities often requires different techniques, such as factoring or using graphical methods.

Absolute Value Inequalities:

Inequalities involving absolute values require careful consideration of the definition of absolute value. They often lead to compound inequalities.

Practical Applications of Linear Inequalities

Linear inequalities are not just abstract mathematical concepts; they have numerous practical applications in various fields:

• Budgeting: Determining how much money can be spent on different items while staying within a budget.
• Resource Allocation: Optimizing the allocation of resources (time, materials, etc.) subject to constraints.
• Production Planning: Determining the optimal production levels of different products given resource limitations and demand.
• Engineering: Designing structures or systems that meet certain specifications and constraints.
• Economics: Modeling economic relationships and predicting outcomes.

Conclusion

Solving linear inequalities, such as (2/3)x - (1/5) > 1, involves applying fundamental algebraic principles to isolate the variable. Understanding the steps involved, interpreting the solution set, and verifying the solution are crucial for accurate and confident problem-solving. This foundational knowledge extends to more advanced mathematical concepts and has widespread applications in various fields. Remember to always check your solution by substituting a value from the solution set back into the original inequality to ensure it satisfies the condition. This meticulous approach will guarantee accurate results and build a strong foundation in your understanding of linear inequalities.