**Solve the Inequality: A Step-by-Step Guide to Simplify and Find Solutions**

In the world of mathematics, inequalities are like puzzles that require careful analysis and strategic moves to solve. Tackling an inequality like 38 + 4x ≥ 3 – 7x can be a daunting task, but with the right approach, you can simplify it and find the solutions with ease.

Imagine yourself as a detective tasked with solving a mystery. Inequalities present challenges that require you to examine each side of the equation, uncover hidden relationships, and uncover the truth hidden within the mathematical symbols.

To solve the given inequality, we’ll embark on a step-by-step journey, breaking down the problem into manageable pieces and applying mathematical principles to simplify and isolate the variable.

**Subtracting Constants:**

- Begin by subtracting 38 from both sides of the inequality.
- This operation gives us 4x ≥ -41 – 7x.

**Combining Like Terms:**

- On the right side of the inequality, combine the constants -41 and -7x.
- This results in 4x ≥ -41 – 7x, which can be simplified further.

**Adding Variables:**

- Add 7x to both sides of the inequality.
- We get 4x + 7x ≥ -41 – 7x + 7x.

**Simplifying:**

- Combine the like terms 4x and 7x on the left side.
- Simplify further to get 11x ≥ -41.

**Dividing by a Coefficient:**

- Divide both sides of the inequality by 11, a positive coefficient.
- This gives us x ≥ -41/11.

Therefore, the solution to the inequality 38 + 4x ≥ 3 – 7x is x ≥ -41/11. This means that any value of x that is greater than or equal to -41/11 will satisfy the inequality.

Remember, solving inequalities is a skill that requires patience, practice, and a systematic approach. With persistence and a willingness to break down the problem into smaller steps, you can conquer any inequality that comes your way.

## Solving the Inequality: 3x + 4 ≤ 7 + 3x

**Introduction:**

In mathematics, inequalities are mathematical expressions that compare two values or expressions using symbols such as <, >, ≤, and ≥. These symbols represent “less than,” “greater than,” “less than or equal to,” and “greater than or equal to,” respectively. Solving inequalities involves isolating the variable on one side of the inequality symbol and obtaining the solution set, which consists of all the values of the variable that satisfy the inequality.

**Simplifying the Inequality:**

The given inequality is 3x + 4 ≤ 7 + 3x. To simplify this inequality, let’s isolate the variable term 3x on one side of the inequality symbol.

**Step 1: Subtract 3x from Both Sides:**

3x + 4 – 3x ≤ 7 + 3x – 3x

4 ≤ 7

**Step 2: Conclusion:**

Since 4 is always less than or equal to 7, the inequality holds true for all values of x. Therefore, the solution set is all real numbers, which can be expressed as:

x ∈ R

**Graphical Representation:**

The solution set of the inequality x ∈ R can be represented graphically on a number line as follows:

```
<center>
<img src="https://tse1.mm.bing.net/th?q=+Graphical+Representation" alt="Graphical Representation">
</center>
```

The entire number line is shaded, indicating that all values of x satisfy the inequality.

**Conclusion:**

The solution to the inequality 3x + 4 ≤ 7 + 3x is x ∈ R, which means that all real numbers satisfy the inequality. This can be seen graphically as the entire number line being shaded.

**FAQs:**

**What is an inequality?**

An inequality is a mathematical expression that compares two values or expressions using symbols such as <, >, ≤, and ≥.

**What is the solution set of an inequality?**

The solution set of an inequality consists of all the values of the variable that satisfy the inequality.

**How do you solve an inequality?**

To solve an inequality, isolate the variable term on one side of the inequality symbol and obtain the solution set.

**What is the solution set of the inequality 3x + 4 ≤ 7 + 3x?**

The solution set of the inequality 3x + 4 ≤ 7 + 3x is x ∈ R, which means that all real numbers satisfy the inequality.

**How can you represent the solution set of an inequality graphically?**

The solution set of an inequality can be represented graphically on a number line, where the solution set is shaded to indicate the values of the variable that satisfy the inequality.

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