Solving Inequalities: -8 - 7m > 6 - 9m And 10m - 9 > 4 + 11m

Nov 5, 2025 by Admin 61 views

Hey guys! Today, we're diving into the world of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like >, <, ≥, and ≤. Unlike equations, which have a single solution, inequalities often have a range of solutions. We're going to tackle two specific inequalities in this article: -8 - 7m > 6 - 9m and 10m - 9 > 4 + 11m. So, grab your thinking caps, and let's get started!

Understanding Inequalities

Before we jump into solving these inequalities, let's quickly recap the basics. Think of inequalities as a balancing act, but instead of finding the exact point of equilibrium, we're looking for a range where one side is "greater than," "less than," "greater than or equal to," or "less than or equal to" the other side. The key difference from equations is that multiplying or dividing both sides of an inequality by a negative number flips the direction of the inequality sign. It's crucial to remember this rule to avoid common mistakes. Understanding the basic principles of inequalities is crucial before tackling more complex problems. This foundation ensures that you can confidently manipulate and solve inequalities, interpret the solutions correctly, and apply these skills in various mathematical and real-world contexts. Mastering inequalities involves not only knowing the rules but also understanding the underlying logic and how they differ from equations. This deeper understanding allows you to approach problems more flexibly and accurately.

Key Concepts in Inequalities

Inequality Symbols: These symbols dictate the relationship between two expressions. The primary symbols include:
- ```
  	**: Greater than**, indicating that the value on one side is larger than the value on the other side.
```
- ```
  	**: Less than**, showing that the value on one side is smaller than the value on the other side.
```
- ≥: Greater than or equal to**, signifying that the value on one side is either larger than or equal to the value on the other side.
- ≤: Less than or equal to**, indicating that the value on one side is either smaller than or equal to the value on the other side.

Basic Operations: Just like equations, we can add, subtract, multiply, and divide both sides of an inequality to isolate the variable. However, there's a critical rule to remember:

  	**Multiplying or Dividing by a Negative Number**: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if you have -2x > 4, dividing both sides by -2 gives x < -2.

Solution Sets: Inequalities often have a range of solutions, rather than a single value. This range is called the solution set, and it can be represented in various ways:

  	**Interval Notation**: A way to express the solution set using intervals. For example, x > 3 can be written as (3, ∞).

  	**Graphing on a Number Line**: Visual representation of the solution set on a number line. This helps to easily see the range of values that satisfy the inequality.

Compound Inequalities: These involve two or more inequalities joined by "and" or "or."

  	**"And" Inequalities**: The solution must satisfy both inequalities. For example, 2 < x < 5 means x is greater than 2 and less than 5.

  	**"Or" Inequalities**: The solution must satisfy at least one of the inequalities. For example, x < 1 or x > 3 means x is either less than 1 or greater than 3.

Properties of Inequalities: These are fundamental rules that allow us to manipulate inequalities while preserving their validity. Some key properties include:

  	**Addition Property**: Adding the same number to both sides of an inequality does not change the inequality.

  	**Subtraction Property**: Subtracting the same number from both sides of an inequality does not change the inequality.

  	**Multiplication Property**: Multiplying both sides by the same positive number does not change the inequality. If multiplying by a negative number, the inequality sign must be reversed.

  	**Division Property**: Dividing both sides by the same positive number does not change the inequality. If dividing by a negative number, the inequality sign must be reversed.

Understanding these key concepts is crucial for solving various types of inequality problems, from simple linear inequalities to more complex compound inequalities. Each concept plays a vital role in the process of finding and expressing the solution set. The properties of inequalities, in particular, provide the rules for manipulating and simplifying inequalities while preserving their validity, making them essential tools in the problem-solving process. Grasping these basics makes tackling complex problems much more manageable and ensures a solid foundation in algebra.

Solving -8 - 7m > 6 - 9m

Let's break down the first inequality: -8 - 7m > 6 - 9m. Our goal is to isolate 'm' on one side of the inequality. Here’s how we can do it:

Combine 'm' terms: To start, let's get all the 'm' terms on one side. We can add 9m to both sides of the inequality:
```
-8 - 7m + 9m > 6 - 9m + 9m
-8 + 2m > 6
```
Isolate the term with 'm': Next, we want to isolate the term with 'm'. We can do this by adding 8 to both sides:
```
-8 + 2m + 8 > 6 + 8
2m > 14
```
Solve for 'm': Now, we need to solve for 'm'. We can divide both sides by 2:
```
2m / 2 > 14 / 2
m > 7
```

So, the solution to the first inequality is m > 7. This means that any value of 'm' greater than 7 will satisfy the inequality. It's like saying,