Solve The Inequality: ${ 7 - 3(6c + 14) \ \textgreater \ 5(1 - 2c) }$

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that two mathematical expressions are not equal, but rather one is greater than, less than, or equal to the other. In this article, we will focus on solving the inequality ${ 7 - 3(6c + 14) \ \textgreater \ 5(1 - 2c) }$, which involves algebraic expressions and variables.

Understanding the Inequality

Before we dive into solving the inequality, let's first understand what it means. The given inequality is ${ 7 - 3(6c + 14) \ \textgreater \ 5(1 - 2c) }$. This means that the expression on the left-hand side, $7 - 3(6c + 14)$, is greater than the expression on the right-hand side, $5(1 - 2c)$.

Distributing and Simplifying

To solve the inequality, we need to distribute and simplify both sides of the inequality. Let's start with the left-hand side:

$${ 7 - 3(6c + 14) }$$

Using the distributive property, we can rewrite this as:

$${ 7 - 18c - 42 }$$

Combining like terms, we get:

$${ -35 - 18c }$$

Now, let's simplify the right-hand side:

$${ 5(1 - 2c) }$$

Using the distributive property, we can rewrite this as:

$${ 5 - 10c }$$

Combining Like Terms

Now that we have simplified both sides of the inequality, we can combine like terms:

$${ -35 - 18c \ \textgreater \ 5 - 10c }$$

Isolating the Variable

To isolate the variable $c$, we need to get all the terms with $c$ on one side of the inequality. Let's add $10c$ to both sides of the inequality:

$${ -35 - 18c + 10c \ \textgreater \ 5 - 10c + 10c }$$

Simplifying, we get:

$${ -35 - 8c \ \textgreater \ 5 }$$

Solving for $c$

Now that we have isolated the variable $c$, we can solve for $c$. Let's add $35$ to both sides of the inequality:

$${ -35 - 8c + 35 \ \textgreater \ 5 + 35 }$$

Simplifying, we get:

$${ -8c \ \textgreater \ 40 }$$

Dividing by a Negative Number

To solve for $c$, we need to divide both sides of the inequality by $-8$. However, when we divide by a negative number, the direction of the inequality sign changes. So, we get:

$${ c \ \textless \ -5 }$$

Conclusion

In this article, we solved the inequality ${ 7 - 3(6c + 14) \ \textgreater \ 5(1 - 2c) }$. We used algebraic expressions and variables to simplify and isolate the variable $c$. The final solution is $c \ \textless \ -5$, which means that the value of $c$ must be less than $-5$ to satisfy the inequality.

Tips and Tricks

• When solving inequalities, it's essential to remember that the direction of the inequality sign changes when we divide by a negative number.
• Always simplify and combine like terms to make the inequality easier to solve.
• Use the distributive property to expand expressions and make them easier to work with.

Real-World Applications

Inequalities have numerous real-world applications, including:

• Finance: Inequalities are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Inequalities are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Final Thoughts

Solving inequalities is a crucial skill in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to simplify and combine like terms, use the distributive property, and be mindful of the direction of the inequality sign when dividing by a negative number. With practice and patience, you'll become proficient in solving inequalities and tackle even the most challenging problems.

Introduction

In our previous article, we solved the inequality ${ 7 - 3(6c + 14) \ \textgreater \ 5(1 - 2c) }$. In this article, we will answer some of the most frequently asked questions about solving inequalities.

Q&A

Q: What is an inequality?

A: An inequality is a statement that two mathematical expressions are not equal, but rather one is greater than, less than, or equal to the other.

Q: How do I know which direction to write the inequality sign?

A: The direction of the inequality sign depends on the operation being performed. For example, if you are adding or subtracting a number, the inequality sign remains the same. However, if you are multiplying or dividing by a negative number, the direction of the inequality sign changes.

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 $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Set each factor equal to zero and solve for the variable.
3. Use a number line or a graph to determine the intervals where the inequality is true.

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

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x > 2$ or $x < 3$. A non-strict inequality, on the other hand, is an inequality that is written with a non-strict inequality sign, such as $x \geq 2$ or $x \leq 3$.

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

A: To graph an inequality on a number line, you can use the following steps:

1. Draw a number line and mark the point that is equal to the value in the inequality.
2. Use a closed circle to indicate the point that is included in the inequality.
3. Use an open circle to indicate the point that is not included in the inequality.
4. Shade the region to the left or right of the point, depending on the direction of the inequality sign.

Q: What is the difference between a linear programming problem and a quadratic programming problem?

A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to a set of linear constraints. A quadratic programming problem, on the other hand, is a problem that involves maximizing or minimizing a quadratic function subject to a set of linear constraints.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, you can use the following steps:

1. Write the objective function and the constraints in standard form.
2. Graph the constraints on a coordinate plane.
3. Find the feasible region, which is the region where all the constraints are satisfied.
4. Find the optimal solution, which is the point in the feasible region that maximizes or minimizes the objective function.

Conclusion

In this article, we answered some of the most frequently asked questions about solving inequalities. We covered topics such as linear inequalities, quadratic inequalities, strict inequalities, non-strict inequalities, graphing inequalities on a number line, and linear programming problems. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy.

Tips and Tricks

• Always read the problem carefully and understand what is being asked.
• Use a number line or a graph to visualize the inequality and determine the intervals where the inequality is true.
• Use the distributive property to expand expressions and make them easier to work with.
• Be mindful of the direction of the inequality sign when dividing by a negative number.

Real-World Applications

Inequalities have numerous real-world applications, including:

• Finance: Inequalities are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Inequalities are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Final Thoughts

Solving inequalities is a crucial skill in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to read the problem carefully, use a number line or a graph to visualize the inequality, and be mindful of the direction of the inequality sign when dividing by a negative number. With practice and patience, you'll become proficient in solving inequalities and tackle even the most challenging problems.