Solve The Linear Inequality For K K K . Graph Your Answer On The Number Line, And Then Write The Solution In Inequality Notation. 6 − 5 K + 6 + 6 K \textgreater 6 − 2 K 6 - 5k + 6 + 6k \ \textgreater \ 6 - 2k 6 − 5 K + 6 + 6 K \textgreater 6 − 2 K Show Your Work Here:Hint: To Add Inequalities ($\ \textless

Mar 12, 2025 by ADMIN 311 views

Solving Linear Inequalities: A Step-by-Step Guide

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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 a linear inequality for $k$ and graphing the solution on a number line. We will also write the solution in inequality notation.

The Given Inequality

The given inequality is:

6 - 5k + 6 + 6k \ \textgreater \ 6 - 2k

Step 1: Simplify the Inequality

To simplify the inequality, we need to combine like terms on both sides of the inequality.

6 - 5k + 6 + 6k \ \textgreater \ 6 - 2k

12 + k \ \textgreater \ 6 - 2k

Step 2: Add 2k to Both Sides

To isolate the variable $k$ , we need to add $2k$ to both sides of the inequality.

12 + k + 2k \ \textgreater \ 6 - 2k + 2k

12 + 3k \ \textgreater \ 6

Step 3: Subtract 12 from Both Sides

To further isolate the variable $k$ , we need to subtract 12 from both sides of the inequality.

12 + 3k - 12 \ \textgreater \ 6 - 12

3k \ \textgreater \ -6

Step 4: Divide Both Sides by 3

To solve for $k$ , we need to divide both sides of the inequality by 3.

\frac{3k}{3} \ \textgreater \ \frac{-6}{3}

k \ \textgreater \ -2

Step 5: Graph the Solution on a Number Line

To graph the solution on a number line, we need to draw a line that represents the inequality $k \ \textgreater \ -2$ . We will use an open circle to indicate that $k$ is greater than $-2$ , but not equal to it.

Step 6: Write the Solution in Inequality Notation

The solution to the inequality is $k \ \textgreater \ -2$ . This can be written in inequality notation as:

k \in (-\infty, -2)

Conclusion

In this article, we solved a linear inequality for $k$ and graphed the solution on a number line. We also wrote the solution in inequality notation. The solution to the inequality is $k \ \textgreater \ -2$ , which can be written in inequality notation as $k \in (-\infty, -2)$ .

Frequently Asked Questions

Q: What is a linear inequality?

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

References

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \ \textless \ cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Q: How do I solve a linear inequality?

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

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

A: To graph a linear inequality on a number line, you need to draw a line that represents the inequality. If the inequality is of the form $x \ \textless \ a$ , you will draw an open circle at $a$ and shade the region to the left of the line. If the inequality is of the form $x \ \textgreater \ a$ , you will draw an open circle at $a$ and shade the region to the right of the line.

Q: How do I write a linear inequality in interval notation?

A: To write a linear inequality in interval notation, you need to identify the values of $x$ that satisfy the inequality. If the inequality is of the form $x \ \textless \ a$ , you will write the solution as $(-\infty, a)$ . If the inequality is of the form $x \ \textgreater \ a$ , you will write the solution as $(a, \infty)$ .

Q: Can I solve a linear inequality by using a calculator?

A: Yes, you can solve a linear inequality by using a calculator. You can use a graphing calculator to graph the inequality and find the solution. Alternatively, you can use a calculator to solve the inequality algebraically.

Q: How do I check my solution to a linear inequality?

A: To check your solution to a linear inequality, you need to plug the solution back into the original inequality and verify that it is true. If the solution satisfies the inequality, then it is a valid solution.

Q: Can I solve a system of linear inequalities?

A: Yes, you can solve a system of linear inequalities. To solve a system of linear inequalities, you need to find the solution that satisfies all of the inequalities in the system. You can use a variety of methods to solve a system of linear inequalities, including graphing, substitution, and elimination.

Q: How do I graph a system of linear inequalities?

A: To graph a system of linear inequalities, you need to graph each inequality in the system on a number line. You will then identify the region that satisfies all of the inequalities in the system.

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

A: Yes, you can use a calculator to solve a system of linear inequalities. You can use a graphing calculator to graph the inequalities and find the solution.

Q: How do I write a system of linear inequalities in matrix form?

A: To write a system of linear inequalities in matrix form, you need to identify the coefficients of the variables in the inequalities. You will then write the coefficients in a matrix and use the matrix to solve the system.

Q: Can I use a computer program to solve a system of linear inequalities?

A: Yes, you can use a computer program to solve a system of linear inequalities. You can use a variety of computer programs, including MATLAB and Python, to solve a system of linear inequalities.

Conclusion

In this article, we have discussed some common questions and answers related to solving linear inequalities. We have covered topics such as graphing linear inequalities, writing linear inequalities in interval notation, and solving systems of linear inequalities. We have also discussed how to use calculators and computer programs to solve linear inequalities.