Solving –4(X + 3) ≤ –2 – 2x: Which Number Line?

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When it comes to solving inequalities, understanding number lines can be a valuable tool. In this case, we are tasked with determining which number line represents the solution set for the inequality –4(X + 3) ≤ –2 – 2x. Exploring the relationship between the given expression and the number line will allow us to visualize the range of values that satisfy this inequality. By carefully analyzing the equation and its components, we can navigate through the complexities of inequalities and unveil the solution set on the number line. So, let's delve into this problem and uncover the answer.


Introduction

In this article, we will discuss the solution set for the inequality –4(X + 3) ≤ –2 – 2x and determine which number line represents this solution set. We will analyze the given inequality step by step and find the correct representation of the solution set.

Understanding the Inequality

To begin, let's break down the given inequality –4(X + 3) ≤ –2 – 2x. This expression involves both variables (X and x) and constants (-4, -2, and 3). We have a combination of addition, subtraction, and multiplication operations in the equation.

Simplifying the Inequality

To simplify the inequality, we can start by distributing the -4 to both terms inside the parentheses: -4 * X and -4 * 3. This results in -4X - 12 ≤ -2 - 2x.

Combining Like Terms

Next, we can combine like terms on both sides of the inequality. On the left side, we have -4X, and on the right side, we have -2 - 2x. Therefore, the simplified inequality becomes -4X - 12 ≤ -2 - 2x.

Bringing Variables to One Side

To bring the variables (X and x) to one side of the inequality, we can add 2x to both sides. This gives us -4X + 2x - 12 ≤ -2.

Combining Variables and Constants

Now, we can combine the variables and constants on the left side of the inequality: -4X + 2x - 12. However, to combine the variables, we need to have the same variable term. Thus, we can rewrite the expression as -4X + 2x = -2x - 4X.

Combining Like Terms Again

After combining the variables, we get -6X - 12 ≤ -2.

Isolating the Variable

To isolate the variable on one side of the inequality, we can add 12 to both sides: -6X - 12 + 12 ≤ -2 + 12. This simplifies to -6X ≤ 10.

Dividing by a Negative Number

Since there is a negative coefficient (-6) attached to the variable (X), we need to divide both sides by -6. However, when dividing by a negative number, the direction of the inequality symbol flips. Therefore, we get X ≥ -10/6 or X ≥ -5/3.

Representing the Solution Set

Now that we have determined the value of X, we can represent the solution set on a number line. The solution set for the inequality X ≥ -5/3 includes all values greater than or equal to -5/3. This means we shade the number line to the right of -5/3 and include -5/3 as well.

Conclusion

In conclusion, the solution set for the inequality –4(X + 3) ≤ –2 – 2x is X ≥ -5/3. This means that any value of X greater than or equal to -5/3 will satisfy the inequality. When representing this solution set on a number line, we shade the region to the right of -5/3, including -5/3. By understanding the steps involved in solving the inequality, we can accurately determine the solution set and represent it visually.


Introduction

When examining an inequality, it is essential to determine which number line accurately represents the solution set. In this case, we will analyze the inequality –4(X + 3) ≤ –2 – 2x and determine the appropriate number line representation.

Inequality Simplification

To begin, let's simplify the given inequality –4(X + 3) ≤ –2 – 2x. We can do this by distributing the –4 to the terms within the parentheses, resulting in -4X - 12 ≤ -2 - 2x.

Combining Like Terms

Next, we can further simplify the inequality by combining like terms. This means combining the terms with the variable X and the terms without variables. Thus, the inequality becomes -4X + 2x ≤ -2 + 12.

Variable Isolation

To effectively identify the solution set, we need to isolate the variable on one side of the inequality. In this case, we can isolate the variable by subtracting 2x from both sides, resulting in -4X - 2x ≤ -2 + 12 - 2x.

Further Simplification

After isolating the variable, we can simplify the inequality further. By combining like terms again, we get -6X ≤ 10 - 2x.

Determining the Direction of the Inequality

Understanding the direction of the inequality is crucial in representing the solution set. In this case, the inequality sign is ≤, indicating that the solution set includes values that are less than or equal to the right-hand side of the inequality.

Inferring the Solution Set on a Number Line

Now, let's apply our knowledge of the inequality direction and plot the solution set accurately on a number line. Since the solution set includes values that are less than or equal to the right-hand side of the inequality, we will plot the solution set to the left of the obtained representation.

Identifying Boundary Points

To determine the boundary points of the solution set, we need to find the values of X that make the inequality true. In this case, the boundary points occur when -6X = 10 - 2x. By solving this equation, we can find the specific values of X that define the boundaries of the solution set.

Analyzing Open and Closed Circles

When representing the boundary points on the number line, it is crucial to determine whether they should be represented as open or closed circles. An open circle indicates that the value is not included in the solution set, while a closed circle represents that the value is included.

Final Solution Set Representation

After determining the boundary points and understanding whether they should be represented as open or closed circles, we can plot the solution set on the number line. Additionally, we will use appropriate notation, such as brackets or parentheses, to accurately represent the inequality.In conclusion, by examining the given inequality –4(X + 3) ≤ –2 – 2x and following the steps outlined above, we can determine the appropriate number line representation for the solution set. It is crucial to simplify the inequality, isolate the variable, analyze the direction of the inequality, locate the boundary points, and accurately plot the solution set on the number line.

Which Number Line Represents The Solution Set For The Inequality –4(X + 3) ≤ –2 – 2x?

Explanation of the Inequality and Solution Set

In order to determine which number line represents the solution set for the inequality –4(X + 3) ≤ –2 – 2x, we must first understand the inequality itself.

The given inequality is –4(X + 3) ≤ –2 – 2x. Let's simplify it.

  1. First, distribute -4 to the terms inside the parentheses:

    2. -4X - 12 ≤ -2 - 2x

  2. Next, combine like terms:

    3. -4X + 2x - 12 ≤ -2

  3. Then, simplify further:

    4. -2X - 12 ≤ -2

  4. To isolate the variable, add 12 to both sides:

    5. -2X ≤ 10

  5. Finally, divide both sides by -2. Remember, when dividing by a negative number, we must flip the inequality sign:

    6. X ≥ -5

Therefore, the solution set for the inequality is X ≥ -5.

Representation on a Number Line

In order to represent the solution set on a number line, we need to plot the values of X that satisfy the inequality X ≥ -5.

  • Start by drawing a number line with zero in the middle.
  • Mark a point at -5 and shade everything to the right of it, including -5.
  • This shaded region represents all the values of X that are greater than or equal to -5, which is the solution set for the given inequality.

Table Information

To further understand the solution set, we can create a table of values that satisfy the inequality. Let's substitute a few values into the inequality and observe the results:

X –4(X + 3) –2 – 2x Satisfies Inequality?
-6 12 10 No
-5 8 8 Yes
-4 4 6 Yes
0 -12 -2 No

From the table, we can see that when X is equal to or greater than -5, the inequality is satisfied. Any value less than -5 does not satisfy the inequality.

Therefore, the number line representing the solution set for the inequality –4(X + 3) ≤ –2 – 2x is the one where all values to the right of -5 are shaded.


Thank you for visiting our blog and taking the time to read our article on solving the inequality –4(X + 3) ≤ –2 – 2x. We hope that the information provided has been helpful in understanding how to find the solution set for this particular inequality. In this closing message, we would like to summarize the key points discussed in the article and highlight the importance of understanding and applying inequalities in mathematics.

In the article, we began by explaining the given inequality –4(X + 3) ≤ –2 – 2x. We demonstrated step-by-step how to simplify the inequality by distributing the –4 to both terms inside the parentheses. This led us to –4x – 12 ≤ –2 – 2x. By combining like terms and isolating the variable, we obtained 2x ≤ 10. Finally, we divided both sides of the inequality by 2, resulting in x ≤ 5. This means that any value of x less than or equal to 5 will satisfy the given inequality.

Understanding and solving inequalities is an essential skill in mathematics. Inequalities help us express relationships between variables and determine the range of values that make an equation true. They are commonly used in various fields such as economics, physics, and engineering. Mastering the concept of inequalities allows us to make informed decisions based on mathematical models and analyze real-world situations more accurately.

In conclusion, we would like to emphasize the significance of grasping the concepts and techniques involved in solving inequalities. The ability to solve inequalities enables us to interpret and solve a wide range of problems in different disciplines. We hope that this article has served as a useful guide in understanding how to find the solution set for the inequality –4(X + 3) ≤ –2 – 2x. If you have any further questions or would like to explore more topics related to inequalities, please feel free to browse our blog for more informative articles. Thank you once again for visiting, and we look forward to sharing more valuable content with you in the future.


Which Number Line Represents The Solution Set For The Inequality –4(X + 3) ≤ –2 – 2x?

1. What is the given inequality?

The given inequality is –4(X + 3) ≤ –2 – 2x.

2. How can we solve this inequality?

To solve this inequality, we need to simplify and isolate the variable on one side of the inequality symbol.

Simplifying steps:

  1. Distribute -4 to terms inside the parentheses: -4X - 12 ≤ -2 - 2x
  2. Combine like terms: -4X + 2x ≤ -2 + 12
  3. Combine x terms: -2X ≤ 10
  4. Divide by -2 (note: when dividing by a negative number, the inequality sign flips): X ≥ -5

3. How do we represent the solution set on a number line?

To represent the solution set on a number line, we use a closed or open circle to indicate the boundary point, and an arrow to show the direction of the solution set.

Number line representation:

We represent the solution set X ≥ -5 on a number line as follows:

  • Draw a number line with -5 as a reference point.
  • Place a closed or filled circle on -5 to represent the inclusive boundary point.
  • Draw an arrow extending towards the right side of the number line, indicating that all values greater than or equal to -5 are solutions to the inequality.

Visually, the number line will look something like this:

             -5--------------------------->