**Simplify and Solve: The Quest for Inequality Solutions**

Do you find yourself grappling with complex mathematical expressions, particularly inequalities, that leave you feeling perplexed? If so, you’re not alone. Understanding how to simplify and solve inequalities can be a daunting task, especially when faced with expressions like 38 + 4x + 3 + 7 – 3x. But fear not, because this guide will break down the process into manageable steps, leaving you well-equipped to conquer any inequality that comes your way.

**Unveiling the Layers of Complexity**

Before we delve into solving the inequality, it’s important to recognize the challenges that can arise. The presence of multiple terms with varying coefficients can make it difficult to visualize the overall structure of the expression. Additionally, the combination of addition and subtraction can create uncertainty in determining which operations should be performed first.

**Simplifying the Expression: A Step-by-Step Guide**

To simplify the expression, we’ll start by combining like terms:

```
38 + 4x + 3 + 7 - 3x = (38 + 3 + 7) + (4x - 3x)
48 + x = x + 48
```

**Solving the Inequality**

Now that we have a simplified expression, we can solve the inequality:

```
x + 48 = x + 48
```

Since both sides of the equation are equal, the solution is:

```
x = x
```

This means that any value of x will satisfy the inequality. In other words, there are no restrictions on the value of x.

**Solving the Inequality: 3x + 4x + 3 > 7 – 3x**

**Introduction:**

In mathematical expressions, an inequality represents a relationship between expressions that are not equal but can be compared using symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Solving an inequality involves isolating the variable on one side of the inequality sign while maintaining the inequality relationship.

**Simplifying the Expression:**

- Combine like terms on the left side: 3x + 4x = 7x
- Simplify the expression on the right side: 7 – 3x = 7 – 3x
- The inequality becomes: 7x + 3 > 7 – 3x

**Isolating the Variable Term:**

- Add 3x to both sides to isolate the variable term on the left side: 7x + 3 + 3x > 7 – 3x + 3x
- Simplify: 10x + 3 > 7

**Simplifying the Constant Term:**

- Subtract 3 from both sides: 10x + 3 – 3 > 7 – 3
- Simplify: 10x > 4

**Dividing by the Coefficient of the Variable:**

- Divide both sides by 10, the coefficient of the variable: (10x) / 10 > 4 / 10
- Simplify: x > 0.4

**Conclusion:**

Therefore, the solution to the inequality 3x + 4x + 3 > 7 – 3x is x > 0.4. This means that all values of x greater than 0.4 satisfy the inequality.

**FAQs:**

**What is the first step in solving an inequality?**

- Simplifying the expression and combining like terms.

**What are the symbols used to represent inequalities?**

- >, <, ≥, ≤

**How do you isolate the variable term in an inequality?**

- Add or subtract the same value to both sides of the inequality.

**How do you solve for x in an inequality?**

- Isolate the variable term, simplify the constant term, and divide both sides by the coefficient of the variable.

**What is the solution to the inequality 3x + 4x + 3 > 7 – 3x?**

- x > 0.4

.

Solve,Following,Inequality