Solve And Graph The Solution Set To The Inequality Below. 6 \textless − 3 ( X − 5 ) ≤ 21 6 \ \textless \ -3(x-5) \leq 21 6 \textless − 3 ( X − 5 ) ≤ 21 Drag The Solution Set To The Correct Location On The Number Line.

Mar 12, 2025 by ADMIN 220 views

Solving and Graphing the Solution Set to the Inequality

=====================================================

Introduction

In this article, we will delve into solving and graphing the solution set to the given inequality: $6 \ \textless \ -3(x-5) \leq 21$ . This involves understanding the concept of inequalities, solving for the variable, and graphing the solution set on a number line.

Understanding the Inequality

The given inequality is a compound inequality, which means it consists of two parts: a less-than part and a less-than-or-equal-to part. The inequality is written as $6 \ \textless \ -3(x-5) \leq 21$ . To solve this inequality, we need to isolate the variable $x$ .

Solving the Inequality

To solve the inequality, we will start by isolating the variable $x$ . We can do this by first distributing the negative sign to the terms inside the parentheses:

6 \ \textless \ -3x + 15 \leq 21

Next, we will subtract 15 from both sides of the inequality to get:

-9 \ \textless \ -3x \leq 6

Now, we will divide both sides of the inequality by -3. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality signs:

3 \ \textgreater \ x \geq -2

Graphing the Solution Set

Now that we have solved the inequality, we can graph the solution set on a number line. The solution set consists of all the values of $x$ that satisfy the inequality. In this case, the solution set is all the values of $x$ that are greater than 3 and less than or equal to -2.

To graph the solution set, we will start by drawing a number line and marking the point $x = -2$ with an open circle. This represents the lower bound of the solution set.

Next, we will draw a closed circle at the point $x = 3$ to represent the upper bound of the solution set.

Finally, we will shade the region between the two points to represent the solution set.

Conclusion

In conclusion, solving and graphing the solution set to the inequality $6 \ \textless \ -3(x-5) \leq 21$ involves understanding the concept of inequalities, solving for the variable, and graphing the solution set on a number line. By following the steps outlined in this article, we can solve and graph the solution set to any compound inequality.

Step-by-Step Solution

Here is a step-by-step solution to the inequality:

Distribute the negative sign: $6 \ \textless \ -3x + 15 \leq 21$
Subtract 15 from both sides: $-9 \ \textless \ -3x \leq 6$
Divide both sides by -3: $3 \ \textgreater \ x \geq -2$

Common Mistakes to Avoid

When solving and graphing the solution set to an inequality, there are several common mistakes to avoid:

Not distributing the negative sign: Failing to distribute the negative sign can lead to incorrect solutions.
Not reversing the inequality signs: Failing to reverse the inequality signs when dividing or multiplying by a negative number can lead to incorrect solutions.
Not graphing the solution set correctly: Failing to graph the solution set correctly can lead to incorrect conclusions.

Real-World Applications

Solving and graphing the solution set to an inequality has several real-world applications:

Optimization problems: Inequality solutions can be used to optimize problems in fields such as business, economics, and engineering.
Data analysis: Inequality solutions can be used to analyze data and make informed decisions.
Science and engineering: Inequality solutions can be used to model and solve problems in fields such as physics, chemistry, and biology.

Final Thoughts

Solving and graphing the solution set to an inequality is an important concept in mathematics. By understanding the concept of inequalities, solving for the variable, and graphing the solution set on a number line, we can solve and graph the solution set to any compound inequality. With practice and patience, anyone can master this concept and apply it to real-world problems.

=====================================================

Introduction

In our previous article, we delved into solving and graphing the solution set to the given inequality: $6 \ \textless \ -3(x-5) \leq 21$ . We covered the steps involved in solving the inequality, graphing the solution set, and provided a step-by-step solution. In this article, we will answer some frequently asked questions (FAQs) related to solving and graphing the solution set to an inequality.

Q&A

Q: What is the difference between a less-than inequality and a less-than-or-equal-to inequality?

A: A less-than inequality is denoted by the symbol < and means that the value of the expression is less than the given value. A less-than-or-equal-to inequality is denoted by the symbol ≤ and means that the value of the expression is less than or equal to the given value.

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

A: To solve an inequality with a negative coefficient, you need to multiply both sides of the inequality by the negative coefficient. However, when you multiply or divide an inequality by a negative number, you need to reverse the direction of the inequality signs.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle on a number line represents the endpoint of a solution set, while an open circle represents the endpoint of a solution set that is not included in the solution set.

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

A: To graph the solution set to an inequality on a number line, you need to mark the endpoints of the solution set with either a closed circle or an open circle, depending on whether the endpoint is included in the solution set. Then, you need to shade the region between the endpoints to represent the solution set.

Q: What are some common mistakes to avoid when solving and graphing the solution set to an inequality?

A: Some common mistakes to avoid when solving and graphing the solution set to an inequality include not distributing the negative sign, not reversing the inequality signs when dividing or multiplying by a negative number, and not graphing the solution set correctly.

Q: What are some real-world applications of solving and graphing the solution set to an inequality?

A: Solving and graphing the solution set to an inequality has several real-world applications, including optimization problems, data analysis, and science and engineering.

Additional Tips and Tricks

Tip 1: Use a number line to visualize the solution set

Using a number line can help you visualize the solution set and make it easier to graph.

Tip 2: Check your work

Always check your work by plugging in values from the solution set into the original inequality to make sure they satisfy the inequality.

Tip 3: Use a calculator to check your work

If you're unsure about your work, you can use a calculator to check your solutions.

Conclusion

Frequently Asked Questions (FAQs)

Q: What is the difference between a linear inequality and a nonlinear 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 nonlinear inequality is an inequality that cannot be written in the form ax + b < c.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the solution set to each inequality and then find the intersection of the solution sets.

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