Find the Perfect Number Line for 2x – 6 ≥ 6(X – 2) + 8!

Have you ever wondered which number line represents the solution set for a given inequality? Well, today we will dive into the world of inequalities and explore how to determine the correct number line representation. Our focus will be on the inequality 2x – 6 ≥ 6(X – 2) + 8. Through careful analysis and the use of transition words, we will unravel the mystery and understand the logic behind finding the solution set. So, let's embark on this journey together and uncover the answer!

Introduction

Inequalities are mathematical expressions that compare two quantities. They can be represented on a number line, which helps visualize the solution set. In this article, we will explore an inequality and determine which number line represents its solution set.

The Given Inequality

The given inequality is 2x – 6 ≥ 6(X – 2) + 8. To find the solution set for this inequality, we need to simplify it and express it in a form that can be represented on a number line.

Simplifying the Inequality

Let's start by simplifying the inequality. Distributing 6 to (X – 2) gives us 6X - 12. Therefore, our inequality becomes 2x – 6 ≥ 6X - 12 + 8.

Combining Like Terms

To further simplify, we combine like terms. The right side of the inequality becomes 6X - 4.

Combining Like Terms on the Left Side

Now, let's combine like terms on the left side of the inequality. We have 2x – 6, which cannot be simplified any further.

Bringing Like Terms Together

To bring like terms together, we subtract 6X from both sides of the inequality. This gives us -4x – 6 ≥ -4.

Isolating x

Next, we isolate x by adding 6 to both sides of the inequality. This results in -4x ≥ 2.

Dividing by a Negative Number

Since we have a negative coefficient (-4) in front of x, we need to divide both sides of the inequality by -4. However, when dividing by a negative number, the direction of the inequality sign flips.

Flipping the Inequality

By dividing both sides of the inequality by -4, we get x ≤ -0.5. Therefore, the solution set for the given inequality is all values of x less than or equal to -0.5.

Representing the Solution Set on a Number Line

Now that we have determined the solution set, let's represent it on a number line. We start by marking -0.5 on the number line.

Shading the Number Line

To represent the values less than or equal to -0.5, we shade the portion of the number line to the left of -0.5. This indicates that any value on or to the left of -0.5 satisfies the given inequality.

Conclusion

In conclusion, the solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8 is x ≤ -0.5. This solution set can be represented on a number line by shading the portion to the left of -0.5. Understanding how to represent inequalities on a number line helps visualize and interpret the solution set of an inequality.

Disclaimer:

The following explanation provides a clear understanding of determining the number line representation for the given inequality in a concise and straightforward manner.

Introduction:

In this explanation, we will discuss the number line representation that accurately represents the solution set for the inequality 2x - 6 ≥ 6(x - 2) + 8.

Defining the inequality:

The given inequality, 2x - 6 ≥ 6(x - 2) + 8, involves comparing two expressions involving the variable x.

Simplifying the inequality:

To begin, we need to simplify both sides of the inequality by applying the distributive property and combining like terms.

Expanding the equation:

By distributing 6 to (x - 2) and simplifying, we can rewrite the inequality as 2x - 6 ≥ 6x - 12 + 8.

Combining like terms:

Further simplification by combining like terms gives us 2x - 6 ≥ 6x - 4.

Bringing terms to one side:

To isolate the variable on one side, we can subtract 2x from both sides of the inequality, resulting in -6 ≥ 4x - 4.

Simplifying further:

By adding 4 to both sides of the inequality, we have -2 ≥ 4x.

Dividing by a positive number:

Since the coefficient of x is positive, we can divide both sides by 4 to solve for x and get -1/2 ≥ x.

Identifying the solution set:

The solution set represents the values of x that make the inequality true, so all numbers less than or equal to -1/2 are part of the solution.

Number line representation:

On a number line, we can indicate the solution set by shading the region to the left of -1/2, including the point itself, as all those values satisfy the given inequality.

Which Number Line Represents The Solution Set For The Inequality 2x – 6 ≥ 6(X – 2) + 8?

Story:

Once upon a time, there was a mathematical problem that needed to be solved. The problem was to determine which number line represents the solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8.

To find the solution, we first need to simplify the inequality. By distributing the 6 on the right side of the equation, we get 2x – 6 ≥ 6x – 12 + 8. Simplifying further, we have 2x – 6 ≥ 6x – 4.

Now, we need to isolate the variable x on one side of the inequality. By moving the terms with x to the left side and the constant terms to the right side, we get -4x ≥ -2.

Since we want to isolate x, we need to divide both sides of the inequality by -4. However, we must remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be flipped. Therefore, we have x ≤ 1/2.

Explanation:

The solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8 is represented by a number line where all values of x are less than or equal to 1/2. This means that any value of x that is less than or equal to 1/2 will satisfy the inequality.

Table Information:

Below is a table showing different values of x and whether they satisfy the inequality or not:

From the table, we can see that any value of x that is less than or equal to 1/2 satisfies the inequality, while values greater than 1/2 do not.

Thank you for visiting our blog today! In this article, we have explored the solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8. By analyzing the given inequality, we have determined the number line that represents its solution set. Let's summarize our findings and understand the implications of this solution set.

After simplifying the inequality, we obtained 2x – 6 ≥ 6x – 12 + 8. By combining like terms, we further simplified it to 2x – 6 ≥ 6x – 4. To solve this inequality, we need to isolate the variable x on one side of the inequality sign. By subtracting 2x from both sides, we get -6 ≥ 4x – 4. We then add 4 to both sides to obtain -2 ≥ 4x.

Now, let's focus on interpreting the solution set for this inequality. Since we have -2 ≥ 4x, dividing both sides by 4 gives us -1/2 ≥ x. This means that any value of x less than or equal to -1/2 will satisfy the given inequality. Therefore, the number line representing the solution set would start at -1/2 and extend indefinitely to the left.

In conclusion, the number line representing the solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8 is a line that starts at -1/2 and extends indefinitely to the left. It is important to understand the implications of this solution set when dealing with similar inequalities in the future. We hope this article has provided you with a clear understanding of how to determine the solution set for such inequalities. Thank you for reading!

Which Number Line Represents The Solution Set For The Inequality 2x – 6 ≥ 6(X – 2) + 8?

1. How can I determine the solution set for the given inequality?

In order to find the solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8, you need to solve it step by step. First, simplify both sides of the inequality.

2. How do I simplify the inequality?

Start by distributing the 6 on the right side of the inequality:

• 2x – 6 ≥ 6X - 12 + 8
• 2x – 6 ≥ 6X - 4

Combine like terms on both sides:

• 2x - 6 ≥ 6X - 4
• 2x - 6 - 6X ≥ -4
• -4X - 6 ≥ -4

Next, isolate the variable by moving constant terms to the opposite side:

• -4X - 6 ≥ -4
• -4X - 6 + 6 ≥ -4 + 6
• -4X ≥ 2

Finally, divide both sides of the inequality by -4, remembering to flip the inequality sign when dividing by a negative number:

• -4X ≥ 2
• X ≤ -0.5

3. Which number line represents the solution set for the inequality?

The solution set for the inequality X ≤ -0.5 is represented by a number line where all values to the left of or including -0.5 are part of the solution. The number line should be shaded to the left of -0.5, indicating that any value less than or equal to -0.5 satisfies the inequality.

Here is an example of how the number line would look:

Therefore, the number line that represents the solution set for the inequality 2x – 6 ≥ 6(X – 2) + 8 is the one with all values to the left of or including -0.5 shaded.