Steps to solve x - a > b type inequalities (Grade 7)

📚 Understanding Inequalities Like $x - a > b$

In mathematics, an inequality is a statement that compares two expressions that are not necessarily equal. Unlike equations, which use an equals sign (=), inequalities use symbols like '>', '<', '$\geq$', or '$\leq$' to show the relationship between the expressions. Solving an inequality means finding the range of values for the variable that makes the inequality true.

📜 A Little History

The concept of inequalities has been around for centuries, closely tied to the development of algebra and the need to represent relationships beyond simple equality. Early mathematicians grappled with comparing quantities and understanding ranges of possible solutions, leading to the symbolic notation we use today. While the precise origin is difficult to pinpoint, the use of inequality symbols became more formalized during the development of modern algebraic notation.

🔑 Key Principles for Solving $x - a > b$ Inequalities

Solving inequalities of the form $x - a > b$ is very similar to solving equations. The main goal is to isolate the variable 'x' on one side of the inequality. Here are the steps:

• ➕ Isolate x: ➕ Add 'a' to both sides of the inequality to get 'x' by itself. This maintains the balance, just like in equations.
• 🧮 Simplify: 🧮 Perform the addition on the right side of the inequality to find the range of values for 'x'.
• ✅ Interpret: ✅ The solution tells you all the possible values of 'x' that make the original inequality true.

🪜 Step-by-Step Solution

Let's break down the solution process:

1. Original Inequality: Start with $x - a > b$.
2. Add 'a' to Both Sides: Add 'a' to both sides: $x - a + a > b + a$.
3. Simplify: This simplifies to $x > b + a$.
4. Solution: The solution is $x > b + a$, which means 'x' is any number greater than 'b + a'.

🌍 Real-World Example

Imagine you need to save more than $50 for a new video game. You already have $20 saved. Let 'x' be the amount you still need to save. Then, $x + 20 > 50$. To find out how much more you need to save, subtract 20 from both sides: $x > 50 - 20$, which simplifies to $x > 30$. So, you need to save more than $30.

💡 Tips and Tricks

• 🧠 Think of it like an equation: 🧠 The process is very similar, but remember the direction of the inequality sign!
• 🔄 If you multiply or divide by a negative number: 🔄 Flip the direction of the inequality sign. For example, if $-x > 5$, then $x < -5$.
• 📈 Graphing the Solution: 📈 You can represent the solution on a number line. Use an open circle for '>' or '<' and a closed circle for '$\geq$' or '$\leq$'.

📝 Practice Quiz

Solve the following inequalities:

1. $x - 3 > 7$
2. $x - 5 > 2$
3. $x - 1 > 4$

Answers:

1. $x > 10$
2. $x > 7$
3. $x > 5$

Conclusion

Solving inequalities of the form $x - a > b$ is a fundamental skill in algebra. By understanding the basic principles and practicing regularly, you can master this concept and build a strong foundation for more advanced mathematical topics.