Analyzing the Given Inequality

Oct 13, 2025 by windailys.com 6 views

In mathematics, functions are used to describe the relationship between variables. Two functions, f $}$ and <span class="katex-error" title="ParseError KaTeX parse error: Expected '', got 'EOF' at end of input: " style="color#cc0000">{ g $$, are defined by f(x) = 2x - 1 $}$ and <span class="katex-error" title="ParseError KaTeX parse error: Expected '', got 'EOF' at end of input: " style="color#cc0000">{ g(x) = 4x - 3 $$, where ${$ x $}$ is a real number. These functions can be used to model various real-world situations, such as the cost of goods, the growth of populations, or the movement of objects.

Analyzing the Given Inequality

The given inequality is ${$ f(x+1) - g(x) \leq 0 $}$. To solve this inequality, we need to first find the expressions for ${$ f(x+1) $}$ and ${$ g(x) $}$.

Finding the Expression for f(x+1)

To find the expression for ${$ f(x+1) $}$, we substitute ${$ x+1 $}$ into the function ${$ f(x) = 2x - 1 $}$.

${$ f(x+1) = 2(x+1) - 1 $}$

Expanding the expression, we get:

${$ f(x+1) = 2x + 2 - 1 $}$

Simplifying the expression, we get:

${$ f(x+1) = 2x + 1 $}$

Finding the Expression for g(x)

The expression for ${$ g(x) $}$ is already given as ${$ g(x) = 4x - 3 $}$.

Substituting the Expressions into the Inequality

Now, we substitute the expressions for ${$ f(x+1) $}$ and ${$ g(x) $}$ into the given inequality:

${$ (2x + 1) - (4x - 3) \leq 0 $}$

Simplifying the inequality, we get:

${$ 2x + 1 - 4x + 3 \leq 0 $}$

Combining like terms, we get:

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

Solving the Inequality

To solve the inequality, we need to isolate the variable ${$ x $}$.

First, we subtract 4 from both sides of the inequality:

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

Next, we divide both sides of the inequality by -2. Remember to reverse the direction of the inequality when dividing by a negative number:

${$ x \geq 2 $}$

Conclusion

Therefore, the value of ${$ x $}$ that satisfies the inequality ${$ f(x+1) - g(x) \leq 0 $}$ is ${$ x \geq 2 $}$.

Real-World Applications

The functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ can be used to model various real-world situations, such as:

The cost of goods: If the cost of a good is ${$ f(x) = 2x - 1 $}$ dollars, and the cost of a similar good is ${$ g(x) = 4x - 3 $}$ dollars, then the inequality ${$ f(x+1) - g(x) \leq 0 $}$ can be used to determine the maximum price difference between the two goods.
The growth of populations: If the population of a city is ${$ f(x) = 2x - 1 $}$ people, and the population of a neighboring city is ${$ g(x) = 4x - 3 $}$ people, then the inequality ${$ f(x+1) - g(x) \leq 0 $}$ can be used to determine the maximum population difference between the two cities.
The movement of objects: If the position of an object is ${$ f(x) = 2x - 1 $}$ meters, and the position of a similar object is ${$ g(x) = 4x - 3 $}$ meters, then the inequality ${$ f(x+1) - g(x) \leq 0 $}$ can be used to determine the maximum distance difference between the two objects.

Conclusion

In the previous article, we discussed the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$, and how they can be used to model various real-world situations. In this article, we will answer some frequently asked questions about these functions.

Q: What is the purpose of the functions f(x) and g(x)?

A: The functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ can be used to model various real-world situations, such as the cost of goods, the growth of populations, or the movement of objects.

Q: How do I use the functions f(x) and g(x) to model real-world situations?

A: To use the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ to model real-world situations, you need to substitute the values of the variables into the functions. For example, if you want to model the cost of goods, you can substitute the cost of the goods into the function ${$ f(x) = 2x - 1 $}$.

Q: What is the significance of the inequality f(x+1) - g(x) ≤ 0?

A: The inequality ${$ f(x+1) - g(x) \leq 0 $}$ is used to determine the maximum price difference, population difference, or distance difference between two objects. The value of ${$ x $}$ that satisfies the inequality is ${$ x \geq 2 $}$.

Q: Can I use the functions f(x) and g(x) to model other real-world situations?

A: Yes, you can use the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ to model other real-world situations. For example, you can use them to model the growth of populations, the movement of objects, or the cost of goods.

Q: How do I determine the maximum price difference, population difference, or distance difference between two objects?

A: To determine the maximum price difference, population difference, or distance difference between two objects, you need to substitute the values of the variables into the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$. Then, you need to solve the inequality ${$ f(x+1) - g(x) \leq 0 $}$ to find the value of ${$ x $}$ that satisfies the inequality.

Q: Can I use the functions f(x) and g(x) to model complex real-world situations?

A: Yes, you can use the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ to model complex real-world situations. However, you may need to use more advanced mathematical techniques, such as calculus or differential equations, to solve the problems.

Conclusion

In conclusion, the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ can be used to model various real-world situations. The inequality ${$ f(x+1) - g(x) \leq 0 $}$ can be used to determine the maximum price difference, population difference, or distance difference between two objects. The value of ${$ x $}$ that satisfies the inequality is ${$ x \geq 2 $}$. We hope that this article has helped you to understand the functions ${$ f(x) = 2x - 1 $}$ and ${$ g(x) = 4x - 3 $}$ and how they can be used to model real-world situations.