What Is The Solution To The Inequality $-3x + 7 \ \textgreater \ 1$?

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Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. In this article, we will focus on solving the inequality $-3x + 7 \ \textgreater \ 1$. This type of inequality is known as a linear inequality, and it involves a linear expression on one side of the inequality sign and a constant on the other side. Solving linear inequalities requires a different approach than solving linear equations, and it involves isolating the variable on one side of the inequality sign.

Understanding the Inequality

The given inequality is $-3x + 7 \ \textgreater \ 1$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to subtract 7 from both sides of the inequality, which gives us $-3x \ \textgreater \ -6$. This step is valid because we are performing the same operation on both sides of the inequality sign.

Isolating the Variable

The next step is to isolate the variable $x$ on one side of the inequality sign. To do this, we need to divide both sides of the inequality by -3. However, when we divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, the inequality becomes $x \ \textless \ 2$. This step is valid because we are performing the same operation on both sides of the inequality sign.

Solution to the Inequality

The solution to the inequality $-3x + 7 \ \textgreater \ 1$ is $x \ \textless \ 2$. This means that any value of $x$ that is less than 2 will satisfy the inequality. In other words, the solution set is all real numbers less than 2.

Graphical Representation

To visualize the solution to the inequality, we can graph the related equation $-3x + 7 = 1$. The graph of this equation is a straight line with a slope of -3 and a y-intercept of 7. The solution to the inequality is all the points on one side of the line. In this case, the solution is all the points to the left of the line.

Conclusion

In conclusion, the solution to the inequality $-3x + 7 \ \textgreater \ 1$ is $x \ \textless \ 2$. This means that any value of $x$ that is less than 2 will satisfy the inequality. The solution set is all real numbers less than 2. We can also visualize the solution to the inequality by graphing the related equation and identifying all the points on one side of the line.

Frequently Asked Questions

Q: What is the solution to the inequality $-3x + 7 \ \textgreater \ 1$?

A: The solution to the inequality is $x \ \textless \ 2$.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This involves performing the same operation on both sides of the inequality sign.

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

A: Solving a linear equation involves isolating the variable on one side of the equation sign, while solving a linear inequality involves isolating the variable on one side of the inequality sign and reversing the direction of the inequality sign when dividing both sides by a negative number.

Q: How do I graph the solution to a linear inequality?

A: To graph the solution to a linear inequality, you need to graph the related equation and identify all the points on one side of the line.

Step-by-Step Solution

Step 1: Subtract 7 from both sides of the inequality

Subtracting 7 from both sides of the inequality gives us $-3x \ \textgreater \ -6$.

Step 2: Divide both sides of the inequality by -3

Dividing both sides of the inequality by -3 gives us $x \ \textless \ 2$.

Step 3: Write the solution to the inequality

The solution to the inequality is $x \ \textless \ 2$.

Real-World Applications

Linear inequalities have many real-world applications, including:

• Budgeting: Linear inequalities can be used to model budget constraints and determine the maximum amount of money that can be spent on different items.
• Resource allocation: Linear inequalities can be used to model resource allocation problems and determine the optimal allocation of resources.
• Optimization: Linear inequalities can be used to model optimization problems and determine the optimal solution.

Conclusion

In conclusion, the solution to the inequality $-3x + 7 \ \textgreater \ 1$ is $x \ \textless \ 2$. This means that any value of $x$ that is less than 2 will satisfy the inequality. The solution set is all real numbers less than 2. We can also visualize the solution to the inequality by graphing the related equation and identifying all the points on one side of the line.

Q&A Article

In this article, we will answer some of the most frequently asked questions about solving linear inequalities. Whether you are a student, a teacher, or a professional, this article will provide you with the information you need to solve linear inequalities with confidence.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression on one side of the inequality sign and a constant on the other side. For example, $-3x + 7 \ \textgreater \ 1$ is a linear inequality.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This involves performing the same operation on both sides of the inequality sign.

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

A: Solving a linear equation involves isolating the variable on one side of the equation sign, while solving a linear inequality involves isolating the variable on one side of the inequality sign and reversing the direction of the inequality sign when dividing both sides by a negative number.

Q: How do I graph the solution to a linear inequality?

A: To graph the solution to a linear inequality, you need to graph the related equation and identify all the points on one side of the line.

Q: What is the solution to the inequality $-3x + 7 \ \textgreater \ 1$?

A: The solution to the inequality is $x \ \textless \ 2$. This means that any value of $x$ that is less than 2 will satisfy the inequality.

Q: How do I determine the direction of the inequality sign?

A: To determine the direction of the inequality sign, you need to consider the sign of the coefficient of the variable. If the coefficient is positive, the inequality sign will be in the same direction as the inequality. If the coefficient is negative, the inequality sign will be in the opposite direction.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: No, you cannot use the same steps to solve a linear inequality as you would to solve a linear equation. When solving a linear inequality, you need to isolate the variable on one side of the inequality sign and reverse the direction of the inequality sign when dividing both sides by a negative number.

Q: How do I check my solution to a linear inequality?

A: To check your solution to a linear inequality, you need to plug the value of the variable into the original inequality and determine if the inequality is true or false.

Q: Can I use a calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, you need to be careful when using a calculator to solve a linear inequality, as the calculator may not always give you the correct solution.

Q: How do I graph the solution to a linear inequality on a number line?

A: To graph the solution to a linear inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality and draw a line through the point that represents the solution.

Q: Can I use the same graph to represent the solution to a linear inequality and a linear equation?

A: No, you cannot use the same graph to represent the solution to a linear inequality and a linear equation. The graph of a linear inequality will have a different shape and orientation than the graph of a linear equation.

Q: How do I determine the solution to a linear inequality with multiple variables?

A: To determine the solution to a linear inequality with multiple variables, you need to isolate one of the variables on one side of the inequality sign and then solve for the other variable.

Q: Can I use the same steps to solve a linear inequality with multiple variables as I would to solve a linear inequality with one variable?

A: No, you cannot use the same steps to solve a linear inequality with multiple variables as you would to solve a linear inequality with one variable. When solving a linear inequality with multiple variables, you need to isolate one of the variables on one side of the inequality sign and then solve for the other variable.

Q: How do I check my solution to a linear inequality with multiple variables?

A: To check your solution to a linear inequality with multiple variables, you need to plug the values of the variables into the original inequality and determine if the inequality is true or false.

Q: Can I use a calculator to solve a linear inequality with multiple variables?

A: Yes, you can use a calculator to solve a linear inequality with multiple variables. However, you need to be careful when using a calculator to solve a linear inequality with multiple variables, as the calculator may not always give you the correct solution.

Q: How do I graph the solution to a linear inequality with multiple variables on a number line?

A: To graph the solution to a linear inequality with multiple variables on a number line, you need to plot points on the number line that represent the solution to the inequality and draw lines through the points that represent the solution.

Q: Can I use the same graph to represent the solution to a linear inequality with multiple variables and a linear equation with multiple variables?

A: No, you cannot use the same graph to represent the solution to a linear inequality with multiple variables and a linear equation with multiple variables. The graph of a linear inequality with multiple variables will have a different shape and orientation than the graph of a linear equation with multiple variables.

Conclusion

In conclusion, solving linear inequalities requires a different approach than solving linear equations. When solving a linear inequality, you need to isolate the variable on one side of the inequality sign and reverse the direction of the inequality sign when dividing both sides by a negative number. By following the steps outlined in this article, you will be able to solve linear inequalities with confidence.