Solve The Inequality { -5(x-1) \ \textgreater \ -40$}$, And Graph The Solution On A Number Line.

Mar 13, 2025 by ADMIN 99 views

Solving the Inequality and Graphing the Solution on a Number Line

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Introduction

In this article, we will focus on solving the inequality ${-5(x-1) > -40}$ and graphing the solution on a number line. We will use algebraic methods to isolate the variable and determine the solution set. Additionally, we will use a number line to visualize the solution and understand its meaning.

Understanding the Inequality

The given inequality is ${-5(x-1) > -40}$ . To solve this inequality, we need to isolate the variable x. We can start by distributing the negative 5 to the terms inside the parentheses.

Distributing the Negative 5

${-5(x-1) = -5x + 5}$

Now, the inequality becomes ${-5x + 5 > -40}$ .

Adding 5 to Both Sides

To isolate the term with the variable, we can add 5 to both sides of the inequality.

${-5x + 5 + 5 > -40 + 5}$

This simplifies to ${-5x + 10 > -35}$ .

Subtracting 10 from Both Sides

Next, we can subtract 10 from both sides of the inequality to further isolate the term with the variable.

${-5x + 10 - 10 > -35 - 10}$

This simplifies to ${-5x > -45}$ .

Dividing Both Sides by -5

To solve for x, we can divide both sides of the inequality by -5. However, since we are dividing by a negative number, we need to reverse the direction of the inequality.

${\frac{-5x}{-5} < \frac{-45}{-5}}$

This simplifies to ${x < 9}$ .

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to identify the values of x that satisfy the inequality. Since the inequality is x < 9, we can plot a point at x = 9 and shade the region to the left of the point.

Understanding the Number Line

The number line is a visual representation of the solution set. The point at x = 9 represents the boundary of the solution set. The region to the left of the point represents the values of x that satisfy the inequality.

Interpreting the Number Line

The number line shows that the solution set is all real numbers less than 9. This means that any value of x that is less than 9 will satisfy the inequality.

Conclusion

In this article, we solved the inequality ${-5(x-1) > -40}$ and graphed the solution on a number line. We used algebraic methods to isolate the variable and determine the solution set. Additionally, we used a number line to visualize the solution and understand its meaning. The solution set is all real numbers less than 9, and the number line provides a visual representation of this solution set.

Frequently Asked Questions

Q: What is the solution set of the inequality?

A: The solution set is all real numbers less than 9.

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

A: To graph the solution on a number line, plot a point at x = 9 and shade the region to the left of the point.

Q: What is the meaning of the number line?

A: The number line is a visual representation of the solution set. It shows that the solution set is all real numbers less than 9.

Additional Resources

References

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Introduction

In our previous article, we solved the inequality ${-5(x-1) > -40}$ and graphed the solution on a number line. We used algebraic methods to isolate the variable and determine the solution set. Additionally, we used a number line to visualize the solution and understand its meaning. In this article, we will answer some frequently asked questions about solving inequalities and graphing on a number line.

Q&A

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 > 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 > d, where a, b, c, and d are constants.

Q: How do I solve a linear inequality with a negative coefficient?

A: To solve a linear inequality with a negative coefficient, you can multiply both sides of the inequality by -1. This will change the direction of the inequality.

Q: What is the meaning of the number line?

A: The number line is a visual representation of the solution set. It shows that the solution set is all real numbers less than 9.

Q: How do I graph a solution set on a number line?

A: To graph a solution set on a number line, plot a point at the boundary of the solution set and shade the region to the left or right of the point, depending on the direction of the inequality.

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 >. A non-strict inequality is an inequality that is written with a non-strict symbol, such as ≤ or ≥.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you can graph the solution sets of each inequality on a number line and find the intersection of the two solution sets.

Q: What is the meaning of the intersection of two solution sets?

A: The intersection of two solution sets is the set of all values that satisfy both inequalities.

Examples

Example 1: Solving a Linear Inequality

Solve the inequality 2x + 3 > 5.

Solution

To solve the inequality, we can subtract 3 from both sides:

2x + 3 - 3 > 5 - 3

This simplifies to 2x > 2.

Next, we can divide both sides by 2:

x > 1

Example 2: Graphing a Solution Set

Graph the solution set of the inequality x < 3 on a number line.

Solution

To graph the solution set, we can plot a point at x = 3 and shade the region to the left of the point.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities and graphing on a number line. We discussed the difference between linear and quadratic inequalities, how to solve linear inequalities with negative coefficients, and how to graph solution sets on a number line. We also discussed the meaning of the number line and how to solve systems of linear inequalities.

Frequently Asked Questions

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

Q: How do I solve a linear inequality with a negative coefficient?

A: To solve a linear inequality with a negative coefficient, you can multiply both sides of the inequality by -1. This will change the direction of the inequality.

Q: What is the meaning of the number line?

A: The number line is a visual representation of the solution set. It shows that the solution set is all real numbers less than 9.

Q: How do I graph a solution set on a number line?

A: To graph a solution set on a number line, plot a point at the boundary of the solution set and shade the region to the left or right of the point, depending on the direction of the inequality.