Solve The Inequality: 2 ( 4 + 2 X ) ≥ 5 X + 5 2(4 + 2x) \geq 5x + 5 2 ( 4 + 2 X ) ≥ 5 X + 5 Choose The Correct Solution:A. X ≤ − 2 X \leq -2 X ≤ − 2 B. X ≥ − 2 X \geq -2 X ≥ − 2 C. X ≤ 3 X \leq 3 X ≤ 3 D. X ≥ 3 X \geq 3 X ≥ 3

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Introduction

In this article, we will focus on solving the given inequality $2(4 + 2x) \geq 5x + 5$. This involves simplifying the expression, isolating the variable, and determining the correct solution set. We will use algebraic techniques to solve the inequality and provide the correct solution among the given options.

Step 1: Simplify the Expression

The first step in solving the inequality is to simplify the expression on both sides. We can start by distributing the 2 on the left-hand side:

$$2(4 + 2x) = 8 + 4x$$

So, the inequality becomes:

$$8 + 4x \geq 5x + 5$$

Step 2: Isolate the Variable

Next, we need to isolate the variable x on one side of the inequality. We can start by subtracting 5x from both sides:

$$8 + 4x - 5x \geq 5x - 5x + 5$$

This simplifies to:

$$8 - x \geq 5$$

Step 3: Solve for x

Now, we need to solve for x. We can start by subtracting 8 from both sides:

$$-x \geq -3$$

Step 4: Multiply by -1

To isolate x, we need to multiply both sides by -1. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality:

$$x \leq 3$$

Conclusion

In conclusion, the correct solution to the inequality $2(4 + 2x) \geq 5x + 5$ is $x \leq 3$. This means that any value of x less than or equal to 3 satisfies the inequality.

Discussion

The solution to the inequality $x \leq 3$ means that the value of x can be any real number less than or equal to 3. This includes all integers and fractions less than or equal to 3.

Example

For example, if we substitute x = 2 into the inequality, we get:

$$2(4 + 2(2)) \geq 5(2) + 5$$

Simplifying this expression, we get:

$$2(4 + 4) \geq 10 + 5$$

$$2(8) \geq 15$$

$$16 \geq 15$$

Since 16 is greater than 15, the inequality is true for x = 2.

Comparison with Options

Now, let's compare our solution with the given options:

A. $x \leq -2$ B. $x \geq -2$ C. $x \leq 3$ D. $x \geq 3$

Our solution is $x \leq 3$, which matches option C.

Final Answer

In conclusion, the correct solution to the inequality $2(4 + 2x) \geq 5x + 5$ is $x \leq 3$. This means that any value of x less than or equal to 3 satisfies the inequality.

Frequently Asked Questions

Q: What is the correct solution to the inequality $2(4 + 2x) \geq 5x + 5$?

A: The correct solution is $x \leq 3$.

Q: How do I solve the inequality $2(4 + 2x) \geq 5x + 5$?

A: To solve the inequality, you need to simplify the expression, isolate the variable, and determine the correct solution set.

Q: What is the difference between the inequality $x \leq 3$ and the inequality $x \geq 3$?

A: The inequality $x \leq 3$ means that the value of x can be any real number less than or equal to 3, while the inequality $x \geq 3$ means that the value of x can be any real number greater than or equal to 3.

References

• [1] Algebraic Techniques for Solving Inequalities
• [2] Simplifying Expressions and Isolating Variables
• [3] Solving Linear Inequalities
  

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Q&A: Solving the Inequality $2(4 + 2x) \geq 5x + 5$

Q: What is the correct solution to the inequality $2(4 + 2x) \geq 5x + 5$?

A: The correct solution is $x \leq 3$. This means that any value of x less than or equal to 3 satisfies the inequality.

Q: How do I solve the inequality $2(4 + 2x) \geq 5x + 5$?

A: To solve the inequality, you need to simplify the expression, isolate the variable, and determine the correct solution set. Here's a step-by-step guide:

1. Simplify the expression on both sides of the inequality.
2. Isolate the variable x on one side of the inequality.
3. Solve for x by performing the necessary operations.

Q: What is the difference between the inequality $x \leq 3$ and the inequality $x \geq 3$?

A: The inequality $x \leq 3$ means that the value of x can be any real number less than or equal to 3, while the inequality $x \geq 3$ means that the value of x can be any real number greater than or equal to 3.

Q: How do I determine the correct solution set for the inequality $2(4 + 2x) \geq 5x + 5$?

A: To determine the correct solution set, you need to consider the values of x that satisfy the inequality. In this case, the solution set is $x \leq 3$, which means that any value of x less than or equal to 3 satisfies the inequality.

Q: Can I use the same steps to solve other inequalities?

A: Yes, the steps outlined above can be used to solve other inequalities. However, you may need to adjust the steps depending on the specific inequality you are solving.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

• Not simplifying the expression on both sides of the inequality
• Not isolating the variable x on one side of the inequality
• Not considering the direction of the inequality
• Not checking the solution set for extraneous solutions

Q: How do I check the solution set for extraneous solutions?

A: To check the solution set for extraneous solutions, you need to substitute the values of x into the original inequality and verify that the inequality is true. If the inequality is not true, then the value of x is an extraneous solution.

Q: What is the importance of solving inequalities?

A: Solving inequalities is an important skill in mathematics, as it allows you to determine the solution set for a given inequality. This can be useful in a variety of applications, including science, engineering, and economics.

Frequently Asked Questions

Q: What is the difference between an inequality and an equation?

A: An inequality is a statement that two expressions are not equal, while an equation is a statement that two expressions are equal.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to simplify the expression on both sides of the inequality, isolate the variable x on one side of the inequality, and determine the correct solution set.

Q: What is the solution set for the inequality $x \geq 2$?

A: The solution set for the inequality $x \geq 2$ is all real numbers greater than or equal to 2.

Q: How do I determine the solution set for a quadratic inequality?

A: To determine the solution set for a quadratic inequality, you need to factor the quadratic expression, determine the values of x that satisfy the inequality, and consider the direction of the inequality.

References

• [1] Algebraic Techniques for Solving Inequalities
• [2] Simplifying Expressions and Isolating Variables
• [3] Solving Linear Inequalities
• [4] Solving Quadratic Inequalities
• [5] Determining Solution Sets for Inequalities