Find The Truth Set Of The Inequalities And Represent Your Results On A Number Line.a. 2 Y + 2 ( 2 Y − 1 ) ≤ 6 + 2 ( Y + 5 2y + 2(2y - 1) \leq 6 + 2(y + 5 2 Y + 2 ( 2 Y − 1 ) ≤ 6 + 2 ( Y + 5 ]

Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $2y + 2(2y - 1) \leq 6 + 2(y + 5)$ and represent the results on a number line.

Step 1: Simplify the Inequality

The first step in solving the inequality is to simplify the expression on the left-hand side. We can start by evaluating the expression inside the parentheses:

$2(2y - 1) = 4y - 2$

Now, we can rewrite the inequality as:

$2y + 4y - 2 \leq 6 + 2(y + 5)$

Combine like terms:

$6y - 2 \leq 6 + 2y + 10$

Step 2: Isolate the Variable

Next, we need to isolate the variable y on one side of the inequality. We can start by adding 2 to both sides of the inequality:

$6y - 2 + 2 \leq 6 + 2y + 10 + 2$

This simplifies to:

$6y \leq 18 + 2y + 12$

Combine like terms:

$6y \leq 30 + 2y$

Subtract 2y from both sides:

$4y \leq 30$

Step 3: Solve for y

Now that we have isolated the variable y, we can solve for y by dividing both sides of the inequality by 4:

$y \leq \frac{30}{4}$

This simplifies to:

$y \leq 7.5$

Representing the Results on a Number Line

To represent the results on a number line, we need to draw a line that includes all values of y that satisfy the inequality. Since y is less than or equal to 7.5, we can draw a line that includes all values of y from negative infinity to 7.5.

Conclusion

In this article, we solved the inequality $2y + 2(2y - 1) \leq 6 + 2(y + 5)$ and represented the results on a number line. We simplified the expression on the left-hand side, isolated the variable y, and solved for y. The final result is that y is less than or equal to 7.5.

Key Takeaways

• Simplify the expression on the left-hand side of the inequality.
• Isolate the variable y on one side of the inequality.
• Solve for y by dividing both sides of the inequality by the coefficient of y.
• Represent the results on a number line.

Additional Resources

For more information on solving inequalities, check out the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Frequently Asked Questions

Q: What is the final result of the inequality? A: The final result is that y is less than or equal to 7.5.

Q: How do I represent the results on a number line? A: Draw a line that includes all values of y from negative infinity to 7.5.

Introduction

In our previous article, we solved the inequality $2y + 2(2y - 1) \leq 6 + 2(y + 5)$ and represented the results on a number line. In this article, we will answer some frequently asked questions about solving inequalities.

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 $ax + b \leq c$ or $ax + b \geq c$, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where a, b, and c are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression on the left-hand side of the inequality. If the expression cannot be factored, you can use the quadratic formula to find the solutions to the equation. Then, you can use a number line or a graph to determine the intervals where the inequality is true.

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 $<, >, \neq$. A non-strict inequality is an inequality that is written with a non-strict symbol, such as $\leq, \geq$. For example, the inequality $x < 5$ is a strict inequality, while the inequality $x \leq 5$ is a non-strict inequality.

Q: How do I represent the results of an inequality on a number line?

A: To represent the results of an inequality on a number line, you need to draw a line that includes all values of the variable that satisfy the inequality. If the inequality is a strict inequality, you should draw a line that does not include the endpoint. If the inequality is a non-strict inequality, you should draw a line that includes the endpoint.

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 the left-hand side of the inequality
• Not isolating the variable on one side of the inequality
• Not solving for the variable correctly
• Not representing the results on a number line correctly

Q: How do I check my work when solving an inequality?

A: To check your work when solving an inequality, you should:

• Simplify the expression on the left-hand side of the inequality
• Isolate the variable on one side of the inequality
• Solve for the variable correctly
• Represent the results on a number line correctly

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We discussed the difference between linear and quadratic inequalities, how to solve linear and quadratic inequalities, and how to represent the results on a number line. We also discussed some common mistakes to avoid and how to check your work when solving an inequality.

Key Takeaways

• Linear inequalities can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where a, b, and c are constants.
• Quadratic inequalities can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where a, b, and c are constants.
• To solve a linear inequality, you need to isolate the variable on one side of the inequality.
• To solve a quadratic inequality, you need to factor the quadratic expression on the left-hand side of the inequality.
• To represent the results of an inequality on a number line, you need to draw a line that includes all values of the variable that satisfy the inequality.

Additional Resources

For more information on solving inequalities, check out the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Frequently Asked Questions

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 $ax + b \leq c$ or $ax + b \geq c$, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where a, b, and c are constants.

Q: How do I solve a linear inequality? A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: How do I solve a quadratic inequality? A: To solve a quadratic inequality, you need to factor the quadratic expression on the left-hand side of the inequality. If the expression cannot be factored, you can use the quadratic formula to find the solutions to the equation. Then, you can use a number line or a graph to determine the intervals where the inequality is true.