Solve The Inequality 12p 7 139

Solving the Inequality: 12p + 7 < 139 – A Comprehensive Guide

Solving inequalities is a fundamental concept in algebra, crucial for understanding various mathematical and real-world problems. This comprehensive guide delves into solving the specific inequality 12p + 7 < 139, providing a step-by-step approach, explaining the underlying principles, and exploring related concepts. We'll also discuss the broader implications of solving inequalities and their applications in different fields.

Understanding Inequalities

Before tackling the specific problem, let's refresh our understanding of inequalities. Unlike equations, which signify equality (=), inequalities express a relationship of order between two expressions. The common inequality symbols are:

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

Solving an inequality means finding the range of values for the variable that makes the inequality statement true. The process is similar to solving equations, but with a crucial difference: multiplying or dividing by a negative number reverses the inequality sign.

Step-by-Step Solution of 12p + 7 < 139

Now, let's tackle the inequality 12p + 7 < 139. Our goal is to isolate the variable 'p' to determine the range of values that satisfy the inequality.

Step 1: Subtract 7 from both sides:

Our first step involves isolating the term with 'p'. We achieve this by subtracting 7 from both sides of the inequality:

12p + 7 - 7 < 139 - 7

This simplifies to:

12p < 132

Step 2: Divide both sides by 12:

To isolate 'p', we divide both sides of the inequality by 12:

12p / 12 < 132 / 12

This simplifies to:

p < 11

Therefore, the solution to the inequality 12p + 7 < 139 is p < 11. This means any value of 'p' less than 11 will make the original inequality true.

Representing the Solution

The solution, p < 11, can be represented in several ways:

• Interval Notation: (-∞, 11). This notation indicates that the solution includes all real numbers from negative infinity up to, but not including, 11. The parenthesis signifies that 11 is not included in the solution set.

• Number Line: A number line can visually represent the solution. You would draw a number line, mark 11, and draw an open circle at 11 (because 11 is not included) and shade the region to the left of 11.

• Set-Builder Notation: {p | p ∈ ℝ, p < 11}. This notation reads as "the set of all p such that p is a real number and p is less than 11."

Checking the Solution

It's always a good practice to check the solution. Let's choose a value of 'p' less than 11, say p = 10, and substitute it into the original inequality:

12(10) + 7 < 139

120 + 7 < 139

127 < 139

The inequality holds true. Now let's try a value greater than or equal to 11, say p = 11:

12(11) + 7 < 139

132 + 7 < 139

139 < 139

This is false. Therefore, our solution, p < 11, is correct.

Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have widespread applications in various fields:

• Economics: Analyzing profit margins, supply and demand, and resource allocation often involves inequalities. For example, a company might want to determine the production level (p) that results in a profit (represented by an inequality).

• Physics: Describing the range of possible values for physical quantities, such as velocity, acceleration, or energy, frequently employs inequalities.

• Engineering: Designing structures and systems often involves constraints and limitations represented by inequalities. For instance, the stress on a beam must be less than its yield strength.

• Computer Science: Algorithm analysis and optimization often involve determining the time or space complexity using inequalities.

• Finance: Modeling financial scenarios, such as investment returns or risk management, often uses inequalities to represent constraints and target levels.

Solving More Complex Inequalities

While the example we tackled was relatively straightforward, solving more complex inequalities often involves multiple steps and may include:

• Combining like terms: This involves grouping similar terms together to simplify the inequality.

• Using the distributive property: This property helps simplify expressions with parentheses.

• Dealing with absolute values: Inequalities involving absolute values require careful consideration of the different cases.

• Solving systems of inequalities: These involve finding the region that satisfies multiple inequalities simultaneously. This is often graphically represented.

Conclusion

Solving inequalities is a vital skill in mathematics with numerous real-world applications. Understanding the principles involved, practicing different types of inequalities, and mastering the various methods of representation are crucial for success in this area. Remember, the key is to isolate the variable while carefully managing the inequality symbol, especially when multiplying or dividing by negative numbers. Through diligent practice and a firm grasp of the fundamental principles, you'll be able to confidently solve even the most challenging inequalities. The solution to the inequality 12p + 7 < 139, p < 11, serves as a foundational example illustrating these essential principles. This understanding paves the way for tackling more advanced mathematical concepts and solving problems across diverse fields.