Solve This Inequality 3q 11 8q 99

Solving the Inequality: 3q + 11 > 8q - 99

This article delves into the step-by-step solution of the inequality 3q + 11 > 8q - 99, providing a comprehensive understanding of the process and incorporating SEO best practices for optimal online visibility. We'll cover not only the solution but also related concepts to enhance your understanding of inequalities and their applications.

Understanding Inequalities

Before diving into the solution, let's establish a firm grasp on what inequalities are and how they differ from equations. An inequality is a mathematical statement that compares two expressions using inequality symbols such as:

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)
• ≠ (not equal to)

Unlike equations, which seek to find a specific value(s) that make the equation true, inequalities identify a range of values that satisfy the statement. The solution to an inequality is typically represented as an interval on a number line.

Solving Linear Inequalities

Our inequality, 3q + 11 > 8q - 99, is a linear inequality because the variable 'q' is raised to the power of 1. Solving linear inequalities involves manipulating the inequality to isolate the variable on one side. The key principle to remember is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.

Step-by-Step Solution: 3q + 11 > 8q - 99

Let's break down the solution step-by-step:

Step 1: Combine like terms.

Our goal is to isolate 'q'. Let's start by moving the 'q' terms to one side and the constant terms to the other. We can subtract 3q from both sides:

3q + 11 - 3q > 8q - 99 - 3q

This simplifies to:

11 > 5q - 99

Step 2: Isolate the constant term.

Next, let's move the constant term (-99) to the left side by adding 99 to both sides:

11 + 99 > 5q - 99 + 99

This simplifies to:

110 > 5q

Step 3: Solve for 'q'.

To isolate 'q', we need to divide both sides by 5:

110/5 > 5q/5

This simplifies to:

22 > q

Step 4: Rewrite the solution.

It's generally more conventional to write the variable on the left side. Remember to reverse the inequality symbol when you switch the sides:

q < 22

Therefore, the solution to the inequality 3q + 11 > 8q - 99 is q < 22. This means any value of 'q' less than 22 will satisfy the original inequality.

Representing the Solution on a Number Line

We can visually represent this solution on a number line. We'll use an open circle at 22 because 'q' is strictly less than 22 (it doesn't include 22). The arrow will point to the left, indicating all values less than 22.

    <---o------------------------>
       22

Interval Notation

Another way to represent the solution is using interval notation. In this case, the solution is represented as:

(-∞, 22)

This indicates that the solution includes all real numbers from negative infinity (-∞) up to, but not including, 22.

Checking the Solution

It's always a good practice to check our solution. Let's choose a value of 'q' less than 22, say q = 20. Substitute this value into the original inequality:

3(20) + 11 > 8(20) - 99

60 + 11 > 160 - 99

71 > 61

This statement is true, confirming that our solution, q < 22, is correct.

Applications of Inequalities

Inequalities have a wide range of applications in various fields, including:

• Physics: Describing the range of possible values for physical quantities like speed, temperature, or pressure.
• Engineering: Setting constraints and limits on design parameters.
• Economics: Modeling economic relationships and predicting outcomes.
• Computer Science: Defining search spaces and optimizing algorithms.
• Statistics: Establishing confidence intervals and hypothesis testing.

Further Exploration: Compound Inequalities

Understanding linear inequalities is a foundation for tackling more complex inequality problems. One such example is compound inequalities, which involve more than one inequality connected by "and" or "or". For example:

• q < 22 and q > 10 This means q must be between 10 and 22.
• q < 10 or q > 22 This means q can be any value less than 10 or greater than 22.

Solving compound inequalities involves solving each individual inequality and then combining the solutions based on the connecting word ("and" or "or").

Advanced Inequalities

Beyond linear inequalities, you can encounter quadratic inequalities, polynomial inequalities, and rational inequalities. These require different techniques to solve, often involving graphing or factoring.

Conclusion

Solving the inequality 3q + 11 > 8q - 99 is a straightforward process involving algebraic manipulation. Understanding the fundamental principles of inequalities, such as combining like terms and reversing the inequality symbol when multiplying or dividing by a negative number, is crucial. By following the step-by-step procedure outlined above, you can confidently solve this and similar linear inequalities. Remember to always check your solution to ensure its accuracy. The ability to solve inequalities is a vital skill in various mathematical and scientific fields, highlighting their practical significance beyond theoretical exercises. Mastering this skill opens doors to understanding and tackling more complex mathematical concepts and real-world applications. Continuing to practice and explore different types of inequalities will build a robust understanding of this important mathematical tool.