Solve For X X X And Graph The Solution On The Number Line Below. If Possible, Simplify Your Answer To A Single Inequality. In Case Of No Solution, Leave The Number Line Blank. − 12 ≤ 3 X − 6 -12 \leq 3x - 6 − 12 ≤ 3 X − 6 And $3x - 6 \geq

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities of the form $ax + b \geq c$ and $ax + b \leq c$, where $a$, $b$, and $c$ are constants. We will also learn how to graph the solution on a number line and simplify the answer to a single inequality if possible.

Understanding the Basics

Before we dive into solving linear inequalities, let's understand the basics. A linear inequality is an inequality that can be written in the form $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants. The variable $x$ is the unknown quantity, and the inequality is true for certain values of $x$.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable $x$ on one side of the inequality. We can do this by adding or subtracting the same value to both sides of the inequality. We can also multiply or divide both sides of the inequality by the same non-zero value.

Let's consider the inequality $-12 \leq 3x - 6$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality.

Step 1: Add 6 to both sides of the inequality

Adding 6 to both sides of the inequality gives us:

$$-12 + 6 \leq 3x - 6 + 6$$

Simplifying the inequality gives us:

$$-6 \leq 3x$$

Step 2: Divide both sides of the inequality by 3

Dividing both sides of the inequality by 3 gives us:

$$\frac{-6}{3} \leq \frac{3x}{3}$$

Simplifying the inequality gives us:

$$-2 \leq x$$

Therefore, the solution to the inequality $-12 \leq 3x - 6$ is $x \geq -2$.

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to plot the points that satisfy the inequality. In this case, the solution is $x \geq -2$, so we need to plot all the points that are greater than or equal to -2.

Here is the number line with the solution graphed:

  -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5
  ---------
  | | | | | | | | | |

As we can see, the solution is all the points that are greater than or equal to -2.

Simplifying the Answer

In some cases, we may be able to simplify the answer to a single inequality. For example, if we have the inequality $x \geq -2$ and $x \leq 2$, we can simplify the answer to $-2 \leq x \leq 2$.

Conclusion

Solving linear inequalities is an important skill for students to master. By following the steps outlined in this article, we can solve linear inequalities of the form $ax + b \geq c$ and $ax + b \leq c$, where $a$, $b$, and $c$ are constants. We can also graph the solution on a number line and simplify the answer to a single inequality if possible.

Example 2: Solving a Linear Inequality with No Solution

Let's consider the inequality $3x - 6 \geq -12$ and $3x - 6 \leq -12$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality.

Step 1: Add 6 to both sides of the inequality

Adding 6 to both sides of the inequality gives us:

$$3x - 6 + 6 \geq -12 + 6$$

Simplifying the inequality gives us:

$$3x \geq -6$$

Step 2: Divide both sides of the inequality by 3

Dividing both sides of the inequality by 3 gives us:

$$\frac{3x}{3} \geq \frac{-6}{3}$$

Simplifying the inequality gives us:

$$x \geq -2$$

However, we also have the inequality $3x - 6 \leq -12$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality.

Step 1: Add 6 to both sides of the inequality

Adding 6 to both sides of the inequality gives us:

$$3x - 6 + 6 \leq -12 + 6$$

Simplifying the inequality gives us:

$$3x \leq -6$$

Step 2: Divide both sides of the inequality by 3

Dividing both sides of the inequality by 3 gives us:

$$\frac{3x}{3} \leq \frac{-6}{3}$$

Simplifying the inequality gives us:

$$x \leq -2$$

As we can see, the two inequalities are contradictory, and there is no solution to the system of inequalities.

Here is the number line with the solution graphed:

  -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5
  ---------
  | | | | | | | | | |

As we can see, there is no solution to the system of inequalities.

Conclusion

$$$$$$$$$$

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \geq c$ or $ax + b \leq c$, 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 $x$ on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants. A linear inequality, on the other hand, is an inequality that can be written in the form $ax + b \geq c$ or $ax + b \leq c$.

Q: How do I graph the solution to a linear inequality on a number line?

A: To graph the solution to a linear inequality on a number line, you need to plot the points that satisfy the inequality. If the inequality is of the form $x \geq a$, you need to plot all the points that are greater than or equal to $a$. If the inequality is of the form $x \leq a$, you need to plot all the points that are less than or equal to $a$.

Q: Can I simplify the answer to a linear inequality?

A: Yes, you can simplify the answer to a linear inequality. If you have two inequalities of the form $x \geq a$ and $x \leq b$, you can simplify the answer to $a \leq x \leq b$.

Q: What if I have a system of linear inequalities?

A: If you have a system of linear inequalities, you need to solve each inequality separately and then find the intersection of the solutions. If the solutions do not intersect, then there is no solution to the system of inequalities.

Q: How do I know if a linear inequality has a solution?

A: To determine if a linear inequality has a solution, you need to check if the inequality is true for any value of $x$. If the inequality is true for any value of $x$, then it has a solution. If the inequality is not true for any value of $x$, then it has no solution.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to make sure that the calculator is set to the correct mode (e.g. "solve" or "graph") and that you are entering the correct values.

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

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

• Not isolating the variable $x$ on one side of the inequality
• Not checking if the inequality is true for any value of $x$
• Not simplifying the answer to a linear inequality
• Not graphing the solution to a linear inequality on a number line

Q: How can I practice solving linear inequalities?

A: You can practice solving linear inequalities by working through examples and exercises in a textbook or online resource. You can also try solving linear inequalities on your own and then checking your answers with a calculator or a teacher.

Conclusion

Solving linear inequalities is an important skill for students to master. By following the steps outlined in this article, you can solve linear inequalities of the form $ax + b \geq c$ and $ax + b \leq c$, where $a$, $b$, and $c$ are constants. You can also graph the solution on a number line and simplify the answer to a single inequality if possible.