Solve The Inequality:$\[ -7 + 4x \leq 1 \\]

Feb 28, 2025 by ADMIN 44 views

$Solving the Inequality: $-7 + 4x \leq 1$$

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. They are used to describe the relationship between variables and constants, and are essential in solving various mathematical problems. In this article, we will focus on solving the inequality $-7 + 4x \leq 1$ , which is a linear inequality. We will use algebraic methods to isolate the variable and find the solution set.

Understanding the Inequality

The given inequality is $-7 + 4x \leq 1$ . This means that the expression $-7 + 4x$ is less than or equal to 1. To solve this inequality, we need to isolate the variable $x$ .

Isolating the Variable

To isolate the variable $x$ , we need to get rid of the constant term $-7$ on the left-hand side of the inequality. We can do this by adding 7 to both sides of the inequality.

-7 + 4x ≤ 1
7 + 4x ≤ 1 + 7
4x ≤ 8

Simplifying the Inequality

Now that we have isolated the variable $x$ , we can simplify the inequality by dividing both sides by 4.

4x ≤ 8
x ≤ 8/4
x ≤ 2

Solution Set

The solution set of the inequality $-7 + 4x \leq 1$ is all the values of $x$ that satisfy the inequality. In this case, the solution set is $x \leq 2$ .

Graphical Representation

The solution set can be represented graphically on a number line. The number line is divided into two parts: the part to the left of 2 and the part to the right of 2. The solution set is represented by the part to the left of 2.

Conclusion

In this article, we solved the inequality $-7 + 4x \leq 1$ using algebraic methods. We isolated the variable $x$ by adding 7 to both sides of the inequality and then simplified the inequality by dividing both sides by 4. The solution set of the inequality is $x \leq 2$ . This inequality can be represented graphically on a number line, where the solution set is represented by the part to the left of 2.

Applications of Inequalities

Inequalities have numerous applications in real-life situations. For example, in finance, inequalities are used to calculate interest rates and investment returns. In physics, inequalities are used to describe the motion of objects and the behavior of physical systems. In engineering, inequalities are used to design and optimize systems.

Types of Inequalities

There are several types of inequalities, including:

Linear inequalities: These are inequalities that can be written in the form $ax + b \leq c$ or $ax + b \geq c$ , where $a$ , $b$ , and $c$ are constants.
Quadratic inequalities: These are inequalities that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$ , where $a$ , $b$ , and $c$ are constants.
Polynomial inequalities: These are inequalities that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \leq 0$ or $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \geq 0$ , where $a_n$ , $a_{n-1}$ , $\ldots$ , $a_1$ , and $a_0$ are constants.

Solving Inequalities

Solving inequalities involves isolating the variable and finding the solution set. There are several methods for solving inequalities, including:

Adding or subtracting the same value to both sides of the inequality
Multiplying or dividing both sides of the inequality by the same non-zero value
Using algebraic manipulations to isolate the variable

Real-World Applications of Inequalities

Inequalities have numerous real-world applications, including:

Finance: Inequalities are used to calculate interest rates and investment returns.
Physics: Inequalities are used to describe the motion of objects and the behavior of physical systems.
Engineering: Inequalities are used to design and optimize systems.
Economics: Inequalities are used to model economic systems and make predictions about economic trends.

Conclusion

Introduction

In our previous article, we solved the inequality $-7 + 4x \leq 1$ using algebraic methods. We isolated the variable $x$ by adding 7 to both sides of the inequality and then simplified the inequality by dividing both sides by 4. The solution set of the inequality is $x \leq 2$ . In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

Q: What is an inequality?

A: An inequality is a statement that two expressions are not equal. It can be written in the form $a \leq b$ or $a \geq b$ , where $a$ and $b$ are expressions.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable and find the solution set. You can do this by adding or subtracting the same value to both sides of the inequality, multiplying or dividing both sides of the inequality by the same non-zero value, or using algebraic manipulations to isolate the variable.

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 $ax + b \leq c$ or $ax + b \geq c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to identify the solution set and represent it on the number line. For example, if the inequality is $x \leq 2$ , you would represent the solution set as the part of the number line to the left of 2.

Q: What are some real-world applications of inequalities?

A: Inequalities have numerous real-world applications, including finance, physics, engineering, and economics. For example, in finance, inequalities are used to calculate interest rates and investment returns. In physics, inequalities are used to describe the motion of objects and the behavior of physical systems.

Q: How do I use algebraic manipulations to solve an inequality?

A: To use algebraic manipulations to solve an inequality, you need to isolate the variable and find the solution set. You can do this by adding or subtracting the same value to both sides of the inequality, multiplying or dividing both sides of the inequality by the same non-zero value, or using algebraic manipulations to isolate the variable.

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

A: A strict inequality is an inequality that can be written in the form $a < b$ or $a > b$ , where $a$ and $b$ are expressions. A non-strict inequality is an inequality that can be written in the form $a \leq b$ or $a \geq b$ , where $a$ and $b$ are expressions.

Q: How do I use inequalities to model real-world problems?

A: To use inequalities to model real-world problems, you need to identify the variables and constraints of the problem and represent them as inequalities. For example, if you are modeling the cost of a product, you might use an inequality to represent the relationship between the cost and the quantity produced.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We discussed the difference between linear and quadratic inequalities, how to graph an inequality on a number line, and some real-world applications of inequalities. We also discussed how to use algebraic manipulations to solve an inequality and the difference between a strict and non-strict inequality. By understanding these concepts, you can use inequalities to model and solve real-world problems.

Additional Resources

Final Thoughts

Solving inequalities is an essential skill in mathematics and has numerous real-world applications. By understanding the concepts and techniques discussed in this article, you can use inequalities to model and solve real-world problems. Remember to always identify the variables and constraints of the problem and represent them as inequalities. With practice and patience, you can become proficient in solving inequalities and apply them to a wide range of problems.