Solve And Graph The Solution Set For The Inequality: ${ 7x + 4 + X \ \textgreater \ 1 + 3x - 12 }$Select The Correct Choice Below, And If Necessary, Provide The Solution Set.

Mar 8, 2025 by ADMIN 178 views

Solve and Graph the Solution Set for the Inequality: 7x + 4 + x > 1 + 3x - 12

Introduction

In this article, we will learn how to solve and graph the solution set for the given inequality: 7x + 4 + x > 1 + 3x - 12. This involves simplifying the inequality, isolating the variable, and then graphing the solution set on a number line.

Step 1: Simplify the Inequality

The first step in solving the inequality is to simplify it by combining like terms. We can start by combining the x terms on the left-hand side of the inequality:

7x + x > 1 + 3x - 12

Combine the x terms:

8x > 1 + 3x - 12

Step 2: Isolate the Variable

Next, we need to isolate the variable x by getting all the x terms on one side of the inequality. We can do this by subtracting 3x from both sides of the inequality:

8x - 3x > 1 - 12

This simplifies to:

5x > -11

Step 3: Solve for x

Now that we have isolated the variable x, we can solve for x by dividing both sides of the inequality by 5:

x > -11/5

Step 4: Graph the Solution Set

To graph the solution set, we need to plot the number -11/5 on a number line and shade the region to the right of it. This represents all the values of x that satisfy the inequality.

Step 5: Write the Solution Set in Interval Notation

The solution set can be written in interval notation as:

(-∞, -11/5)

This represents all the values of x that are less than -11/5.

Conclusion

In this article, we learned how to solve and graph the solution set for the given inequality: 7x + 4 + x > 1 + 3x - 12. We simplified the inequality, isolated the variable, and then graphed the solution set on a number line. The solution set can be written in interval notation as (-∞, -11/5).

Frequently Asked Questions

Q: What is the solution set for the inequality 7x + 4 + x > 1 + 3x - 12? A: The solution set is (-∞, -11/5).
Q: How do I graph the solution set on a number line? A: To graph the solution set, plot the number -11/5 on a number line and shade the region to the right of it.
Q: What is the meaning of the solution set (-∞, -11/5)? A: The solution set (-∞, -11/5) represents all the values of x that are less than -11/5.

Example Problems

Solve the inequality 2x + 5 > 3x - 2 and graph the solution set on a number line.
Solve the inequality x - 3 > 2x + 1 and graph the solution set on a number line.
Solve the inequality 4x + 2 > 2x + 6 and graph the solution set on a number line.

Tips and Tricks

When solving inequalities, always simplify the inequality by combining like terms.
When isolating the variable, get all the x terms on one side of the inequality.
When graphing the solution set, plot the number on a number line and shade the region to the right of it.

Real-World Applications

Inequalities are used in real-world applications such as finance, economics, and engineering.
Inequalities can be used to model real-world situations such as budgeting, investment, and resource allocation.
Inequalities can be used to solve problems such as finding the maximum or minimum value of a function.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline

Additional Resources

Khan Academy: Inequalities
Mathway: Inequalities
Wolfram Alpha: Inequalities

Note: The above article is a comprehensive guide to solving and graphing the solution set for the given inequality. It includes step-by-step instructions, examples, and real-world applications. The article also includes frequently asked questions, example problems, tips and tricks, and references.

Introduction

In this article, we will answer some frequently asked questions about solving and graphing inequalities. We will cover topics such as simplifying inequalities, isolating variables, and graphing solution sets.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to simplify it by combining like terms. This involves adding or subtracting the same value to both sides of the inequality.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable, get all the x terms on one side of the inequality. This can be done by adding or subtracting the same value to both sides of the inequality.

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 > 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 > d, where a, b, c, and d are constants.

Q: How do I graph the solution set for a linear inequality?

A: To graph the solution set for a linear inequality, plot the number on a number line and shade the region to the right of it (for greater than) or to the left of it (for less than).

Q: What is the meaning of the solution set for a linear inequality?

A: The solution set for a linear inequality represents all the values of x that satisfy the inequality.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, first factor the quadratic expression, if possible. Then, use the factored form to find the values of x that make the expression equal to zero. Finally, use a number line or a graph to determine the solution set.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses the symbols > or <, such as x > 2. A non-strict inequality is an inequality that uses the symbols ≥ or ≤, such as x ≥ 2.

Q: How do I determine the solution set for a system of linear inequalities?

A: To determine the solution set for a system of linear inequalities, graph each inequality on a number line and find the intersection of the solution sets.

Q: What is the meaning of the solution set for a system of linear inequalities?

A: The solution set for a system of linear inequalities represents all the values of x that satisfy all the inequalities in the system.

Q: How do I use inequalities in real-world applications?

A: Inequalities can be used in real-world applications such as finance, economics, and engineering. They can be used to model real-world situations such as budgeting, investment, and resource allocation.

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

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

Not simplifying the inequality before solving it
Not isolating the variable correctly
Not graphing the solution set correctly
Not considering the direction of the inequality

Q: How do I check my work when solving inequalities?

A: To check your work when solving inequalities, plug in a value of x that is in the solution set and verify that the inequality is true. You can also graph the solution set and check that it is correct.

Q: What are some resources for learning more about inequalities?

A: Some resources for learning more about inequalities include:

Khan Academy: Inequalities
Mathway: Inequalities
Wolfram Alpha: Inequalities
Algebra and Trigonometry by Michael Sullivan
College Algebra by James Stewart
Mathematics for the Nonmathematician by Morris Kline