Solve The Inequality And Graph The Solution.$\frac{k}{4} \ \textless \ 2$To Draw A Ray, Plot An Endpoint And Select An Arrow. Select An Endpoint To Change It From Closed To Open. Select The Middle Of The Ray To Delete It.

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Introduction

In mathematics, inequalities are used to compare two or more values. They are an essential part of algebra and are used to solve a wide range of problems. In this article, we will focus on solving and graphing inequalities, specifically the inequality k4 \textless 2\frac{k}{4} \ \textless \ 2. We will break down the steps to solve this inequality and provide a step-by-step guide on how to graph the solution.

Understanding the Inequality

The given inequality is k4 \textless 2\frac{k}{4} \ \textless \ 2. This is a linear inequality, which means it can be solved using basic algebraic operations. The inequality states that the value of k4\frac{k}{4} is less than 2.

Step 1: Multiply Both Sides by 4

To solve the inequality, we need to isolate the variable kk. The first step is to multiply both sides of the inequality by 4. This will eliminate the fraction and make it easier to solve.

k4 \textless 2\frac{k}{4} \ \textless \ 2

4×k4 \textless 4×24 \times \frac{k}{4} \ \textless \ 4 \times 2

k \textless 8k \ \textless \ 8

Step 2: Write the Solution in Interval Notation

The solution to the inequality is k \textless 8k \ \textless \ 8. This can be written in interval notation as (−∞,8)(-\infty, 8).

Graphing the Solution

To graph the solution, we need to plot the endpoint of the interval on a number line. Since the inequality is k \textless 8k \ \textless \ 8, we will plot the point 8 on the number line and draw an open circle around it.

Step 2.1: Plot the Endpoint

Plot the point 8 on the number line.

2.2: Draw an Open Circle

Draw an open circle around the point 8.

2.3: Draw a Ray

Draw a ray starting from the point 8 and extending to the left. This represents the solution to the inequality.

Conclusion

In this article, we solved the inequality k4 \textless 2\frac{k}{4} \ \textless \ 2 and graphed the solution. We broke down the steps to solve the inequality and provided a step-by-step guide on how to graph the solution. The solution to the inequality is k \textless 8k \ \textless \ 8, which can be written in interval notation as (−∞,8)(-\infty, 8). We also graphed the solution by plotting the endpoint of the interval on a number line and drawing an open circle around it.

Frequently Asked Questions

Q: What is the solution to the inequality k4 \textless 2\frac{k}{4} \ \textless \ 2?

A: The solution to the inequality is k \textless 8k \ \textless \ 8, which can be written in interval notation as (−∞,8)(-\infty, 8).

Q: How do I graph the solution to the inequality?

A: To graph the solution, plot the endpoint of the interval on a number line and draw an open circle around it. Then, draw a ray starting from the point and extending to the left.

Q: What is the difference between a closed and open circle on a number line?

A: A closed circle represents the endpoint of an interval, while an open circle represents the endpoint of an interval that is not included in the solution.

Additional Resources

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

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Introduction

In our previous article, we discussed solving and graphing inequalities, specifically the inequality k4 \textless 2\frac{k}{4} \ \textless \ 2. We broke down the steps to solve the inequality and provided a step-by-step guide on how to graph the solution. 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 \textless cax + b \ \textless \ c, where aa, bb, and cc are constants. A quadratic inequality is an inequality that can be written in the form ax2+bx+c \textless 0ax^2 + bx + c \ \textless \ 0, where aa, bb, and cc are constants.

Q: How do I solve a linear inequality with a variable on both sides?

A: To solve a linear inequality with a variable on both sides, 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.

Q: What is the difference between a closed and open circle on a number line?

A: A closed circle represents the endpoint of an interval, while an open circle represents the endpoint of an interval that is not included in the solution.

Q: How do I graph a solution to a linear inequality?

A: To graph a solution to a linear inequality, you need to plot the endpoint of the interval on a number line and draw a ray starting from the endpoint and extending to the left or right.

Q: What is the difference between a horizontal and vertical line on a number line?

A: A horizontal line represents a constant value, while a vertical line represents a variable value.

Q: How do I determine the direction of a ray on a number line?

A: To determine the direction of a ray on a number line, you need to look at the inequality sign. If the inequality sign is less than (<), the ray should extend to the left. If the inequality sign is greater than (>), the ray should extend to the right.

Q: What is the difference between a solution and a graph?

A: A solution is the set of values that satisfy an inequality, while a graph is a visual representation of the solution.

Q: How do I check if a solution is correct?

A: To check if a solution is correct, you need to plug the solution back into the original inequality and check if it is true.

Conclusion

In this article, we answered some frequently asked questions about solving and graphing inequalities. We discussed the difference between linear and quadratic inequalities, how to solve linear inequalities with a variable on both sides, and how to graph a solution to a linear inequality. We also discussed the difference between a closed and open circle on a number line, how to determine the direction of a ray on a number line, and how to check if a solution is correct.

Additional Resources

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