Solve The Inequality:$\[ 4x - 2000 + 1.5x \geq 3000 \\]

Feb 27, 2025 by ADMIN 56 views

Solving Inequalities: A Step-by-Step Guide to Solving the Inequality 4x - 2000 + 1.5x ≥ 3000

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that two mathematical expressions are not equal, but one is either greater than or less than the other. In this article, we will focus on solving the inequality 4x - 2000 + 1.5x ≥ 3000. We will break down the solution into step-by-step instructions, making it easy to understand and follow.

Before we dive into solving the inequality, let's first understand what it means. The given inequality is 4x - 2000 + 1.5x ≥ 3000. This means that the expression 4x - 2000 + 1.5x is greater than or equal to 3000. Our goal is to find the values of x that satisfy this inequality.

Step 1: Combine Like Terms

The first step in solving the inequality is to combine like terms. In this case, we have two terms with x, which are 4x and 1.5x. We can combine these terms by adding their coefficients.

${ 4x + 1.5x = 5.5x }$

So, the inequality becomes:

${ 5.5x - 2000 ≥ 3000 }$

Step 2: Add 2000 to Both Sides

The next step is to add 2000 to both sides of the inequality. This will help us isolate the term with x.

${ 5.5x - 2000 + 2000 ≥ 3000 + 2000 }$

${ 5.5x ≥ 5000 }$

Step 3: Divide Both Sides by 5.5

Now, we need to divide both sides of the inequality by 5.5 to solve for x.

${ \frac{5.5x}{5.5} ≥ \frac{5000}{5.5} }$

${ x ≥ \frac{5000}{5.5} }$

Step 4: Simplify the Right-Hand Side

To simplify the right-hand side, we can divide 5000 by 5.5.

${ x ≥ 909.09 }$

In conclusion, we have solved the inequality 4x - 2000 + 1.5x ≥ 3000. By following the step-by-step instructions, we have found that x ≥ 909.09. This means that any value of x greater than or equal to 909.09 will satisfy the inequality.

Solving inequalities has numerous real-world applications. In economics, inequalities are used to model the relationship between variables such as supply and demand. In engineering, inequalities are used to design and optimize systems. In finance, inequalities are used to model the behavior of financial markets.

Here are some tips and tricks to help you solve inequalities:

Always start by combining like terms.
Add or subtract the same value to both sides of the inequality.
Multiply or divide both sides of the inequality by the same non-zero value.
Use inverse operations to isolate the term with the variable.

Here are some common mistakes to avoid when solving inequalities:

Not combining like terms.
Not adding or subtracting the same value to both sides of the inequality.
Not multiplying or dividing both sides of the inequality by the same non-zero value.
Not using inverse operations to isolate the term with the variable.

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the step-by-step instructions and avoiding common mistakes, you can solve inequalities with ease. Remember to always combine like terms, add or subtract the same value to both sides of the inequality, multiply or divide both sides of the inequality by the same non-zero value, and use inverse operations to isolate the term with the variable.
Solving Inequalities: A Q&A Guide

In our previous article, we discussed how to solve the inequality 4x - 2000 + 1.5x ≥ 3000. We broke down the solution into step-by-step instructions, making it easy to understand and follow. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that two mathematical expressions are not equal, but one is either greater than or less than the other.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities are inequalities that can be written in the form ax + b ≥ c or ax + b ≤ c, where a, b, and c are constants. Quadratic inequalities are inequalities that can be written in the form ax^2 + bx + c ≥ 0 or ax^2 + bx + c ≤ 0, where a, b, and c are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow these steps:

Combine like terms.
Add or subtract the same value to both sides of the inequality.
Multiply or divide both sides of the inequality by the same non-zero value.
Use inverse operations to isolate the term with the variable.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to follow these steps:

Factor the quadratic expression.
Set each factor equal to zero and solve for x.
Use a number line to determine the intervals where the inequality is true.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: The main difference between a linear inequality and a quadratic inequality is the form of the inequality. A linear inequality can be written in the form ax + b ≥ c or ax + b ≤ c, while a quadratic inequality can be written in the form ax^2 + bx + c ≥ 0 or ax^2 + bx + c ≤ 0.

Q: How do I graph an inequality?

A: To graph an inequality, you need to follow these steps:

Plot the boundary line.
Test a point in each interval to determine which intervals are true.
Shade the intervals that are true.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

Not combining like terms.
Not adding or subtracting the same value to both sides of the inequality.
Not multiplying or dividing both sides of the inequality by the same non-zero value.
Not using inverse operations to isolate the term with the variable.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug your solution back into the original inequality and verify that it is true.

If you are struggling with solving inequalities, here are some additional resources that may help:

Khan Academy: Inequalities
Mathway: Inequalities
Wolfram Alpha: Inequalities