Solve The Inequality:${ 13 - (17 - 9v) \ \textgreater \ 4(v - 1) }$

Mar 11, 2025 by ADMIN 71 views

Solving the Inequality: A Step-by-Step Guide

=====================================================

In this article, we will delve into the world of inequalities and explore a specific problem that requires us to isolate the variable and find the solution set. The given inequality is $13 - (17 - 9v) \ \textgreater \ 4(v - 1)$ , and our goal is to solve for $v$ .

Understanding the Inequality

Before we begin solving the inequality, it's essential to understand the concept of inequalities and how they differ from equations. An inequality is a statement that compares two expressions using a mathematical symbol, such as $>$ , $<$ , $\geq$ , or $\leq$ . In this case, we have a greater-than inequality, which means we need to find the values of $v$ that make the expression on the left-hand side greater than the expression on the right-hand side.

Distributing and Simplifying

To solve the inequality, we need to start by distributing and simplifying the expressions on both sides. We can begin by distributing the negative sign inside the parentheses on the left-hand side:

13 - (17 - 9v) \ \textgreater \ 4(v - 1)

13 - 17 + 9v \ \textgreater \ 4v - 4

-4 + 9v \ \textgreater \ 4v - 4

Next, we can add 4 to both sides to get rid of the negative term:

9v + 4 \ \textgreater \ 4v

Isolating the Variable

Now that we have simplified the inequality, we can isolate the variable $v$ by subtracting $4v$ from both sides:

9v - 4v + 4 \ \textgreater \ 4v - 4v

5v + 4 \ \textgreater \ 0

Solving for $v$

To solve for $v$ , we need to isolate the variable on one side of the inequality. We can do this by subtracting 4 from both sides:

5v \ \textgreater \ -4

Next, we can divide both sides by 5 to get:

v \ \textgreater \ -\frac{4}{5}

Conclusion

In this article, we solved the inequality $13 - (17 - 9v) \ \textgreater \ 4(v - 1)$ by distributing and simplifying the expressions, isolating the variable, and solving for $v$ . We found that the solution set is $v \ \textgreater \ -\frac{4}{5}$ . This means that any value of $v$ greater than $-\frac{4}{5}$ will satisfy the inequality.

Example Use Cases

The solution to this inequality has various applications in real-world scenarios. For instance, in finance, the inequality can be used to determine the minimum return on investment required to achieve a certain level of profitability. In engineering, the inequality can be used to design systems that meet specific performance criteria.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

Always start by distributing and simplifying the expressions on both sides.
Isolate the variable by subtracting or adding terms to both sides.
Be careful when dividing both sides by a negative number, as it will flip the direction of the inequality.
Check your solution by plugging in values to ensure that they satisfy the original inequality.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Here are some common pitfalls to avoid:

Failing to distribute and simplify the expressions on both sides.
Not isolating the variable correctly.
Flipping the direction of the inequality when dividing both sides by a negative number.
Not checking the solution by plugging in values.

Final Thoughts

Solving inequalities requires a combination of algebraic skills and logical thinking. By following the steps outlined in this article, you can master the art of solving inequalities and apply it to real-world problems. Remember to always check your solution and be mindful of common mistakes to ensure that you arrive at the correct answer.

=====================================

In the previous article, we explored the concept of solving inequalities and walked through a step-by-step guide to solving the inequality $13 - (17 - 9v) \ \textgreater \ 4(v - 1)$ . In this article, we will address some of the most frequently asked questions about solving inequalities.

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value or values of the variable that make the equation true, whereas solving an inequality involves finding the set of values of the variable that make the inequality true.

Q: How do I know which direction to flip the inequality when dividing both sides by a negative number?

A: When dividing both sides of an inequality by a negative number, you need to flip the direction of the inequality. For example, if you have $x > 5$ and you divide both sides by $-2$ , the resulting inequality would be $x < -\frac{5}{2}$ .

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. For example, if you have $x > 5$ and you add $3$ to both sides, the resulting inequality would be $x + 3 > 8$ .

Q: Can I multiply or divide both sides of an inequality by the same value?

A: Yes, you can multiply or divide both sides of an inequality by the same value, but be careful not to flip the direction of the inequality if you are dividing by a negative number.

Q: How do I know if an inequality is true or false?

A: To determine if an inequality is true or false, you need to test a value of the variable that satisfies the inequality. If the value satisfies the inequality, then the inequality is true. If the value does not satisfy the inequality, then the inequality is false.

Q: Can I use the same steps to solve a compound inequality as I would to solve a single inequality?

A: Yes, you can use the same steps to solve a compound inequality as you would to solve a single inequality. However, you need to be careful to handle the multiple inequalities correctly.

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

A: To graph an inequality on a number line, you need to plot a point on the number line that satisfies the inequality and then shade the region to the left or right of the point, depending on the direction of the inequality.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality, but be careful to enter the correct values and to check the results carefully.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug in a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use the same steps to solve a linear inequality as I would to solve a quadratic inequality?

A: Yes, you can use the same steps to solve a linear inequality as you would to solve a quadratic inequality. However, you need to be careful to handle the different types of inequalities correctly.

Q: How do I know if an inequality is linear or quadratic?

A: To determine if an inequality is linear or quadratic, you need to examine the terms on both sides of the inequality. If the terms are linear, then the inequality is linear. If the terms are quadratic, then the inequality is quadratic.

Q: Can I use the same steps to solve an inequality with absolute value as I would to solve a linear inequality?

A: Yes, you can use the same steps to solve an inequality with absolute value as you would to solve a linear inequality. However, you need to be careful to handle the absolute value correctly.

Q: How do I know if an inequality with absolute value is true or false?

A: To determine if an inequality with absolute value is true or false, you need to test a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use a graphing calculator to solve an inequality with absolute value?

A: Yes, you can use a graphing calculator to solve an inequality with absolute value, but be careful to enter the correct values and to check the results carefully.

Q: How do I check my solution to an inequality with absolute value?

A: To check your solution to an inequality with absolute value, you need to plug in a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use the same steps to solve a system of inequalities as I would to solve a single inequality?

A: Yes, you can use the same steps to solve a system of inequalities as you would to solve a single inequality. However, you need to be careful to handle the multiple inequalities correctly.

Q: How do I know if a system of inequalities is consistent or inconsistent?

A: To determine if a system of inequalities is consistent or inconsistent, you need to examine the inequalities and determine if they have a common solution.

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

A: Yes, you can use a graphing calculator to solve a system of inequalities, but be careful to enter the correct values and to check the results carefully.

Q: How do I check my solution to a system of inequalities?

A: To check your solution to a system of inequalities, you need to plug in a value of the variable that satisfies the inequalities and verify that it satisfies the original inequalities.

Q: Can I use the same steps to solve a linear programming problem as I would to solve a system of inequalities?

A: Yes, you can use the same steps to solve a linear programming problem as you would to solve a system of inequalities. However, you need to be careful to handle the different types of inequalities correctly.

Q: How do I know if a linear programming problem is feasible or infeasible?

A: To determine if a linear programming problem is feasible or infeasible, you need to examine the inequalities and determine if they have a common solution.

Q: Can I use a graphing calculator to solve a linear programming problem?

A: Yes, you can use a graphing calculator to solve a linear programming problem, but be careful to enter the correct values and to check the results carefully.

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

A: To check your solution to a linear programming problem, you need to plug in a value of the variable that satisfies the inequalities and verify that it satisfies the original inequalities.

Q: Can I use the same steps to solve a quadratic inequality as I would to solve a linear inequality?

A: Yes, you can use the same steps to solve a quadratic inequality as you would to solve a linear inequality. However, you need to be careful to handle the different types of inequalities correctly.

Q: How do I know if a quadratic inequality is true or false?

A: To determine if a quadratic inequality is true or false, you need to test a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use a graphing calculator to solve a quadratic inequality?

A: Yes, you can use a graphing calculator to solve a quadratic inequality, but be careful to enter the correct values and to check the results carefully.

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

A: To check your solution to a quadratic inequality, you need to plug in a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use the same steps to solve a rational inequality as I would to solve a linear inequality?

A: Yes, you can use the same steps to solve a rational inequality as you would to solve a linear inequality. However, you need to be careful to handle the different types of inequalities correctly.

Q: How do I know if a rational inequality is true or false?

A: To determine if a rational inequality is true or false, you need to test a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use a graphing calculator to solve a rational inequality?

A: Yes, you can use a graphing calculator to solve a rational inequality, but be careful to enter the correct values and to check the results carefully.

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

A: To check your solution to a rational inequality, you need to plug in a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use the same steps to solve a polynomial inequality as I would to solve a linear inequality?

A: Yes, you can use the same steps to solve a polynomial inequality as you would to solve a linear inequality. However, you need to be careful to handle the different types of inequalities correctly.

Q: How do I know if a polynomial inequality is true or false?

A: To determine if a polynomial inequality is true or false, you need to test a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.

Q: Can I use a graphing calculator to solve a polynomial inequality?

A: Yes, you can use a graphing calculator to solve a polynomial inequality, but be careful to enter the correct values and to check the results carefully.

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

A: To check your solution to a polynomial inequality, you need to plug in a value of the variable that satisfies the inequality and verify that it satisfies the original inequality.