Solve The Inequality: $\[ -20 \leq 4x \\]

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation. We will use the inequality ${-20 \leq 4x}$ as an example to demonstrate the steps involved in solving linear inequalities.

What are Linear Inequalities?

Linear inequalities are inequalities that can be written in the form of a linear equation. They are of the form $ax + b \leq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear inequalities can be classified into three types:

• Greater than or equal to: $ax + b \geq c$
• Less than or equal to: $ax + b \leq c$
• Greater than: $ax + b > c$
• Less than: $ax + b < c$

**Solving the Inequality ${-20 \leq 4x}$

To solve the inequality ${-20 \leq 4x}$, we need to isolate the variable $x$. We can do this by dividing both sides of the inequality by $4$. However, we need to be careful when dividing both sides of an inequality by a negative number, as it will change the direction of the inequality.

Step 1: Divide Both Sides by 4

$${-20 \leq 4x}$$

$${\frac{-20}{4} \leq \frac{4x}{4}}$$

$${-5 \leq x}$$

Step 2: Write the Solution in Interval Notation

The solution to the inequality ${-20 \leq 4x}$ can be written in interval notation as $[-5, \infty)$.

Why is it Important to Solve Inequalities?

Solving inequalities is an essential skill in mathematics, as it allows us to compare two expressions and determine the relationship between them. Inequalities are used in various fields, such as economics, finance, and science, to model real-world problems and make predictions.

Real-World Applications of Inequalities

Inequalities have numerous real-world applications, including:

• Economics: Inequalities are used to model the relationship between supply and demand, and to determine the optimal price of a product.
• Finance: Inequalities are used to calculate interest rates, and to determine the minimum amount of money required to invest in a particular asset.
• Science: Inequalities are used to model the behavior of physical systems, such as the motion of objects, and to determine the optimal conditions for a particular process.

Conclusion

Solving inequalities is a crucial skill in mathematics, as it allows us to compare two expressions and determine the relationship between them. In this article, we used the inequality ${-20 \leq 4x}$ as an example to demonstrate the steps involved in solving linear inequalities. We also discussed the importance of solving inequalities and their real-world applications.

Frequently Asked Questions

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

A: A linear equation is an equation that can be written in the form of $ax + b = c$, while a linear inequality is an inequality that can be written in the form of $ax + b \leq c$ or $ax + b \geq c$.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by adding or subtracting the same value from both sides, and then multiplying or dividing both sides by the same value.

Q: What is the significance of solving inequalities?

A: Solving inequalities is essential in mathematics, as it allows us to compare two expressions and determine the relationship between them. Inequalities are used in various fields, such as economics, finance, and science, to model real-world problems and make predictions.

Q: Can you provide more examples of solving inequalities?

A: Yes, here are a few more examples:

• $${2x + 3 \geq 5}$$

• $${x - 2 < 3}$$

• $${4x - 1 \leq 2}$$

Q&A: Frequently Asked Questions about Solving Inequalities

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

A: A linear equation is an equation that can be written in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form of $ax + b \leq c$ or $ax + b \geq c$. The main difference between the two is that a linear equation states that two expressions are equal, while a linear inequality states that one expression is greater than, less than, or equal to another expression.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by adding or subtracting the same value from both sides, and then multiplying or dividing both sides by the same value. The steps involved in solving a linear inequality are:

1. Add or subtract the same value from both sides to isolate the variable.
2. Multiply or divide both sides by the same value to solve for the variable.
3. Write the solution in interval notation.

Q: What is the significance of solving inequalities?

A: Solving inequalities is essential in mathematics, as it allows us to compare two expressions and determine the relationship between them. Inequalities are used in various fields, such as economics, finance, and science, to model real-world problems and make predictions.

Q: Can you provide more examples of solving inequalities?

A: Yes, here are a few more examples:

• $${2x + 3 \geq 5}$$

• $${x - 2 < 3}$$

• $${4x - 1 \leq 2}$$

These examples demonstrate the steps involved in solving linear inequalities, and how to isolate the variable by adding or subtracting the same value from both sides, and then multiplying or dividing both sides by the same value.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as $<$ or $>$, while a non-strict inequality is an inequality that is written with a non-strict symbol, such as $\leq$ or $\geq$. Strict inequalities are used to indicate that one expression is strictly greater than or less than another expression, while non-strict inequalities are used to indicate that one expression is greater than or equal to or less than or equal to another expression.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. The steps involved in graphing a linear inequality are:

1. Graph the corresponding linear equation.
2. Determine the direction of the inequality.
3. Shade the region that satisfies the inequality.

Q: What is the relationship between linear inequalities and linear equations?

A: Linear inequalities and linear equations are closely related. In fact, a linear inequality can be written as a linear equation by adding or subtracting a constant from both sides. For example, the linear inequality $2x + 3 \geq 5$ can be written as the linear equation $2x + 3 - 5 \geq 0$, which simplifies to $2x - 2 \geq 0$.

Q: Can you provide more information about solving systems of linear inequalities?

A: Yes, solving systems of linear inequalities involves solving a system of two or more linear inequalities. The steps involved in solving a system of linear inequalities are:

1. Solve each inequality separately.
2. Graph the corresponding linear equations.
3. Shade the region that satisfies the system of inequalities.

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

A: A linear inequality is an inequality that can be written in the form of $ax + b \leq c$ or $ax + b \geq c$, while a quadratic inequality is an inequality that can be written in the form of $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$. Quadratic inequalities are more complex than linear inequalities and require different techniques to solve.

Q: Can you provide more information about solving quadratic inequalities?

A: Yes, solving quadratic inequalities involves factoring the quadratic expression and then using the factored form to determine the solution. The steps involved in solving a quadratic inequality are:

1. Factor the quadratic expression.
2. Determine the sign of the quadratic expression.
3. Use the factored form to determine the solution.

Conclusion

Solving inequalities is a crucial skill in mathematics, as it allows us to compare two expressions and determine the relationship between them. In this article, we have discussed the basics of solving linear inequalities, including the steps involved in solving a linear inequality, the significance of solving inequalities, and the relationship between linear inequalities and linear equations. We have also provided more information about solving systems of linear inequalities, the difference between a linear inequality and a quadratic inequality, and the steps involved in solving a quadratic inequality.