Two Functions, { F $}$ And { G $}$, Are Defined By $ F X \rightarrow 2x^{-1 }$ And $ G X \rightarrow 4x - 2, }$where { X $ $ Is A Real Number.If { F(x) - G(x) \leq 0 $}$, Find The

Introduction

In mathematics, inequalities are a fundamental concept used to describe relationships between variables. When dealing with two functions, $f(x)$ and $g(x)$, we often need to find the values of $x$ that satisfy certain conditions. In this article, we will explore the problem of finding the values of $x$ that satisfy the inequality $f(x) - g(x) \leq 0$, where $f(x) = 2x^{-1}$ and $g(x) = 4x - 2$.

Understanding the Functions

Before we dive into solving the inequality, let's take a closer look at the two functions involved. The function $f(x) = 2x^{-1}$ is a reciprocal function, which means that it has a vertical asymptote at $x = 0$. This function is also known as the "2/x" function. On the other hand, the function $g(x) = 4x - 2$ is a linear function, which means that it has a constant slope.

The Inequality

Now that we have a good understanding of the two functions, let's take a closer look at the inequality $f(x) - g(x) \leq 0$. To solve this inequality, we need to find the values of $x$ that make the expression $f(x) - g(x)$ less than or equal to zero.

Step 1: Simplify the Expression

To simplify the expression $f(x) - g(x)$, we can substitute the expressions for $f(x)$ and $g(x)$ into the inequality. This gives us:

$$2x^{-1} - (4x - 2) \leq 0$$

Step 2: Combine Like Terms

Next, we can combine like terms in the expression $2x^{-1} - (4x - 2)$. This gives us:

$$2x^{-1} - 4x + 2 \leq 0$$

Step 3: Move All Terms to One Side

To make it easier to solve the inequality, we can move all the terms to one side of the inequality. This gives us:

$$2x^{-1} - 4x + 2 \leq 0$$

$$2x^{-1} - 4x \leq -2$$

Step 4: Factor Out Common Terms

Now that we have moved all the terms to one side of the inequality, we can factor out common terms. In this case, we can factor out a $2$ from the left-hand side of the inequality. This gives us:

$$2(x^{-1} - 2x) \leq -2$$

Step 5: Divide Both Sides by 2

To isolate the expression $x^{-1} - 2x$, we can divide both sides of the inequality by $2$. This gives us:

$$x^{-1} - 2x \leq -1$$

Step 6: Solve for x

Now that we have isolated the expression $x^{-1} - 2x$, we can solve for $x$. To do this, we can multiply both sides of the inequality by $x$, which gives us:

$$1 - 2x^2 \leq -x$$

Step 7: Rearrange the Inequality

To make it easier to solve the inequality, we can rearrange the terms. This gives us:

$$2x^2 + x - 1 \geq 0$$

Step 8: Factor the Quadratic Expression

Now that we have rearranged the terms, we can factor the quadratic expression. This gives us:

$$(2x - 1)(x + 1) \geq 0$$

Step 9: Find the Critical Points

To find the critical points of the inequality, we need to find the values of $x$ that make the expression $(2x - 1)(x + 1)$ equal to zero. This gives us:

$$2x - 1 = 0 \quad \text{or} \quad x + 1 = 0$$

Solving for $x$, we get:

$$x = \frac{1}{2} \quad \text{or} \quad x = -1$$

Step 10: Test the Intervals

Now that we have found the critical points, we can test the intervals to see which ones satisfy the inequality. We can do this by plugging in a value from each interval into the expression $(2x - 1)(x + 1)$.

Interval 1: $(-\infty, -1)$

Let's plug in $x = -2$ into the expression $(2x - 1)(x + 1)$. This gives us:

$$(2(-2) - 1)((-2) + 1) = (-5)(-1) = 5$$

Since the expression is positive, this interval does not satisfy the inequality.

Interval 2: $(-1, \frac{1}{2})$

Let's plug in $x = 0$ into the expression $(2x - 1)(x + 1)$. This gives us:

$$(2(0) - 1)(0 + 1) = (-1)(1) = -1$$

Since the expression is negative, this interval does not satisfy the inequality.

Interval 3: $(\frac{1}{2}, \infty)$

Let's plug in $x = 1$ into the expression $(2x - 1)(x + 1)$. This gives us:

$$(2(1) - 1)((1) + 1) = (1)(2) = 2$$

Since the expression is positive, this interval satisfies the inequality.

Conclusion

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Introduction

In our previous article, we explored the problem of finding the values of $x$ that satisfy the inequality $f(x) - g(x) \leq 0$, where $f(x) = 2x^{-1}$ and $g(x) = 4x - 2$. In this article, we will answer some common questions related to solving inequalities with two functions.

Q: What is the difference between a function and an inequality?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). An inequality, on the other hand, is a statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other.

Q: How do I know which function to use when solving an inequality?

A: When solving an inequality, you need to choose the function that is most relevant to the problem. In this case, we used the function $f(x) = 2x^{-1}$ and $g(x) = 4x - 2$ to solve the inequality $f(x) - g(x) \leq 0$. You can choose any function that is relevant to the problem and use it to solve the inequality.

Q: What is the significance of the critical points in solving an inequality?

A: The critical points are the values of $x$ that make the expression $(2x - 1)(x + 1)$ equal to zero. These points divide the number line into intervals, and we can test each interval to see which ones satisfy the inequality.

Q: How do I know which interval satisfies the inequality?

A: To determine which interval satisfies the inequality, you need to plug in a value from each interval into the expression $(2x - 1)(x + 1)$. If the expression is positive, then the interval satisfies the inequality. If the expression is negative, then the interval does not satisfy the inequality.

Q: Can I use any method to solve an inequality?

A: While there are many methods to solve an inequality, the most common method is to use the critical points to divide the number line into intervals and test each interval to see which ones satisfy the inequality. However, you can also use other methods, such as graphing or using algebraic manipulations, to solve an inequality.

Q: How do I know if an inequality is true or false?

A: To determine if an inequality is true or false, you need to test each interval to see which ones satisfy the inequality. If an interval satisfies the inequality, then the inequality is true. If an interval does not satisfy the inequality, then the inequality is false.

Q: Can I use inequalities to solve real-world problems?

A: Yes, inequalities can be used to solve real-world problems. For example, you can use inequalities to model population growth, optimize resource allocation, or determine the best course of action in a given situation.

Conclusion

In conclusion, solving inequalities with two functions requires a deep understanding of the functions and the inequality itself. By using critical points to divide the number line into intervals and testing each interval to see which ones satisfy the inequality, you can solve inequalities and apply them to real-world problems.

Additional Resources

• Solving Inequalities with Two Functions
• Inequalities in Real-World Applications
• Critical Points and Intervals

Frequently Asked Questions

• Q: What is the difference between a function and an inequality? A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). An inequality, on the other hand, is a statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other.
• Q: How do I know which function to use when solving an inequality? A: When solving an inequality, you need to choose the function that is most relevant to the problem.
• Q: What is the significance of the critical points in solving an inequality? A: The critical points are the values of $x$ that make the expression $(2x - 1)(x + 1)$ equal to zero.
• Q: How do I know which interval satisfies the inequality? A: To determine which interval satisfies the inequality, you need to plug in a value from each interval into the expression $(2x - 1)(x + 1)$.