**Hook:**

Are you struggling to simplify algebraic expressions like “6 less than the product of 4 and x”? Don’t despair! Understanding this concept can unlock a world of mathematical possibilities.

**Pain Points:**

Simplifying algebraic expressions can be challenging, especially when dealing with operations like multiplication and subtraction. Many students encounter difficulties understanding the order of operations and how it affects the simplification process.

**Target:**

By the end of this post, you’ll gain a clear understanding of how to simplify “6 less than the product of 4 and x.” You’ll learn the steps involved and practice with examples to enhance your algebraic skills.

**Main Points:**

- The “product of 4 and x” refers to the multiplication of 4 and x, which can be expressed as 4x.
- “6 less than” means subtracting 6 from the calculated value of 4x.
- The simplified expression becomes 4x – 6.
- Simplifying algebraic expressions involves understanding the order of operations (multiplication before subtraction) and applying it correctly.
- Practicing with various examples helps solidify your understanding of the process.

**Let x Be Any Real Number. Prove That x^2 – 1 ≤ x^2**

**Introduction**

In this article, we will explore the mathematical inequality x^2 – 1 ≤ x^2 for any real number x. We will demonstrate this inequality using algebraic manipulations and provide a geometric interpretation to enhance understanding.

**Derivation of the Inequality**

**Algebraic Manipulation:**

Starting with the left-hand side of the inequality, x^2 – 1, we can rewrite it as follows:

```
x^2 - 1 = (x + 1)(x - 1)
```

Since x can be any real number, at least one of the factors, x + 1 or x – 1, will be non-negative. Therefore, their product, (x + 1)(x – 1), is non-negative.

Adding this non-negative quantity to x^2, which is always non-negative, results in a non-negative quantity:

```
(x^2 - 1) + x^2 = 2x^2 ≥ 0
```

Simplifying the left-hand side gives us:

```
x^2 - 1 ≤ x^2
```

Hence, we have proven the desired inequality.

**Geometric Interpretation:**

Consider a square with side length x. The area of this square is given by x^2. Now, imagine inscribing a circle inside this square, tangent to all four sides. The radius of this circle is x/2.

The area of the circle is given by π(x/2)^2 = (π/4)x^2, which is always less than the area of the square, x^2. This geometric interpretation provides a visual representation of the inequality x^2 – 1 ≤ x^2.

**Conclusion**

We have established that the inequality x^2 – 1 ≤ x^2 holds true for any real number x. This inequality is useful in various mathematical applications, such as calculus and optimization.

**FAQs**

**Q: What is the significance of this inequality?**- A: It provides insight into the relationship between the squares of consecutive integers and is crucial in understanding the behavior of quadratic functions.
**Q: Can the inequality be reversed?**- A: No, it is not possible to reverse the inequality because there exist negative real numbers for which x^2 – 1 > x^2.
**Q: What practical applications does this inequality have?**- A: This inequality is used in proving the concavity of quadratic functions and finding the minimum value of certain functions.

.

Less,Than,Product