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Introduction

In this article, we will delve into the world of inequalities and learn how to solve and graph the given inequality $-12 \leq y - 6$. Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, and graphing the solution involves representing the solution set on a number line.

Understanding the Inequality

The given inequality is $-12 \leq y - 6$. To solve this inequality, we need to isolate the variable $y$ on one side of the inequality sign. We can do this by adding $6$ to both sides of the inequality.

Solving the Inequality

To solve the inequality, we add $6$ to both sides:

$$-12 + 6 \leq y - 6 + 6$$

This simplifies to:

$$-6 \leq y$$

Graphing the Solution

To graph the solution, we need to represent the solution set on a number line. The solution set is all real numbers greater than or equal to $-6$. We can represent this on a number line by drawing a closed circle at $-6$ and shading the region to the right of $-6$.

Step-by-Step Solution

Here's a step-by-step solution to the inequality:

1. Understand the inequality: The given inequality is $-12 \leq y - 6$.
2. Add 6 to both sides: Add $6$ to both sides of the inequality to isolate the variable $y$.
3. Simplify: Simplify the inequality to get $-6 \leq y$.
4. Graph the solution: Represent the solution set on a number line by drawing a closed circle at $-6$ and shading the region to the right of $-6$.

Conclusion

In this article, we learned how to solve and graph the inequality $-12 \leq y - 6$. We added $6$ to both sides of the inequality to isolate the variable $y$, and then represented the solution set on a number line. The solution set is all real numbers greater than or equal to $-6$.

Frequently Asked Questions

• What is the solution to the inequality?: The solution to the inequality is $y \geq -6$.
• How do I graph the solution?: To graph the solution, draw a closed circle at $-6$ and shade the region to the right of $-6$.
• What is the meaning of the inequality?: The inequality $-12 \leq y - 6$ means that $y$ is greater than or equal to $-6$.

Additional Resources

• Inequality basics: Learn the basics of inequalities, including how to solve and graph them.
• Graphing inequalities: Learn how to graph inequalities on a number line.
• Solving inequalities: Learn how to solve inequalities, including how to isolate the variable on one side of the inequality sign.

Final Thoughts

Solving and graphing inequalities is an important skill in mathematics. By understanding how to solve and graph inequalities, you can represent complex mathematical relationships on a number line and make informed decisions. In this article, we learned how to solve and graph the inequality $-12 \leq y - 6$. We added $6$ to both sides of the inequality to isolate the variable $y$, and then represented the solution set on a number line. The solution set is all real numbers greater than or equal to $-6$.

Introduction

In our previous article, we learned how to solve and graph the inequality $-12 \leq y - 6$. We added $6$ to both sides of the inequality to isolate the variable $y$, and then represented the solution set on a number line. The solution set is all real numbers greater than or equal to $-6$. In this article, we will answer some frequently asked questions about solving and graphing 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 $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, 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 sign. You can do this by adding or subtracting the same value to both sides of the inequality.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to represent the solution set on a number line. If the inequality is of the form $x \leq a$, you draw a closed circle at $a$ and shade the region to the left of $a$. If the inequality is of the form $x \geq a$, you draw a closed circle at $a$ and shade the region to the right of $a$.

Q: What is the solution to the inequality $x + 2 \leq 5$?

A: To solve the inequality, you need to isolate the variable $x$ on one side of the inequality sign. You can do this by subtracting $2$ from both sides of the inequality:

$$x + 2 - 2 \leq 5 - 2$$

This simplifies to:

$$x \leq 3$$

Q: How do I graph the solution to the inequality $x + 2 \leq 5$?

A: To graph the solution, you need to represent the solution set on a number line. You draw a closed circle at $3$ and shade the region to the left of $3$.

Q: What is the solution to the inequality $x - 3 \geq 2$?

A: To solve the inequality, you need to isolate the variable $x$ on one side of the inequality sign. You can do this by adding $3$ to both sides of the inequality:

$$x - 3 + 3 \geq 2 + 3$$

This simplifies to:

$$x \geq 5$$

Q: How do I graph the solution to the inequality $x - 3 \geq 2$?

A: To graph the solution, you need to represent the solution set on a number line. You draw a closed circle at $5$ and shade the region to the right of $5$.

Q: Can I have a negative number as a solution to an inequality?

A: Yes, you can have a negative number as a solution to an inequality. For example, the inequality $x \leq -3$ has a solution of $x = -4$, which is a negative number.

Q: Can I have a fraction as a solution to an inequality?

A: Yes, you can have a fraction as a solution to an inequality. For example, the inequality $x \leq \frac{1}{2}$ has a solution of $x = \frac{1}{3}$, which is a fraction.

Conclusion

In this article, we answered some frequently asked questions about solving and graphing inequalities. We learned how to solve and graph linear inequalities, and we saw some examples of how to solve and graph inequalities with negative and fractional solutions. We hope this article has been helpful in answering your questions about solving and graphing inequalities.

Additional Resources

• Inequality basics: Learn the basics of inequalities, including how to solve and graph them.
• Graphing inequalities: Learn how to graph inequalities on a number line.
• Solving inequalities: Learn how to solve inequalities, including how to isolate the variable on one side of the inequality sign.

Final Thoughts

Solving and graphing inequalities is an important skill in mathematics. By understanding how to solve and graph inequalities, you can represent complex mathematical relationships on a number line and make informed decisions. In this article, we answered some frequently asked questions about solving and graphing inequalities, and we hope this article has been helpful in answering your questions.