Solve For $a$.$5(a+16) \ \textgreater \ 20$

Feb 26, 2025 by ADMIN 46 views

$Solve for $a$: $5(a+16) \ \textgreater \ 20$$

Introduction

In mathematics, solving inequalities is a crucial concept that helps us understand the relationship between different variables. In this article, we will focus on solving the inequality $5(a+16) \ \textgreater \ 20$ to find the value of $a$ . This type of problem is commonly encountered in algebra and is an essential skill to master for students and professionals alike.

Understanding the Inequality

The given inequality is $5(a+16) \ \textgreater \ 20$ . To solve for $a$ , we need to isolate the variable $a$ on one side of the inequality. The first step is to distribute the coefficient $5$ to the terms inside the parentheses. This will give us $5a + 80 \ \textgreater \ 20$ .

Distributing the Coefficient

When we distribute the coefficient $5$ to the terms inside the parentheses, we get $5a + 80 \ \textgreater \ 20$ . This is because the coefficient $5$ is multiplied to both the terms $a$ and $16$ inside the parentheses.

Isolating the Variable

Now that we have the inequality in the form $5a + 80 \ \textgreater \ 20$ , we need to isolate the variable $a$ on one side of the inequality. To do this, we will subtract $80$ from both sides of the inequality. This will give us $5a \ \textgreater \ -60$ .

Subtracting 80 from Both Sides

When we subtract $80$ from both sides of the inequality, we get $5a \ \textgreater \ -60$ . This is because the term $80$ is subtracted from both sides of the inequality, resulting in a change in the constant term.

Dividing Both Sides by 5

Now that we have the inequality in the form $5a \ \textgreater \ -60$ , we need to isolate the variable $a$ by dividing both sides of the inequality by $5$ . This will give us $a \ \textgreater \ -12$ .

Dividing Both Sides by 5

When we divide both sides of the inequality by $5$ , we get $a \ \textgreater \ -12$ . This is because the coefficient $5$ is divided out of the inequality, resulting in a change in the constant term.

Conclusion

In conclusion, to solve the inequality $5(a+16) \ \textgreater \ 20$ , we need to isolate the variable $a$ on one side of the inequality. By distributing the coefficient $5$ to the terms inside the parentheses, subtracting $80$ from both sides, and dividing both sides by $5$ , we get the solution $a \ \textgreater \ -12$ . This type of problem is commonly encountered in algebra and is an essential skill to master for students and professionals alike.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that the correct steps are taken.
When distributing coefficients to terms inside parentheses, make sure to multiply the coefficient to each term separately.
When subtracting or adding terms to both sides of an inequality, make sure to change the direction of the inequality sign if necessary.
When dividing both sides of an inequality by a coefficient, make sure to check if the coefficient is positive or negative to determine the direction of the inequality sign.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields such as economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between different variables such as supply and demand. In finance, inequalities can be used to model the relationship between different financial instruments such as stocks and bonds. In engineering, inequalities can be used to model the relationship between different physical quantities such as speed and distance.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid making mistakes such as forgetting to distribute coefficients to terms inside parentheses or forgetting to change the direction of the inequality sign when subtracting or adding terms to both sides.
When dividing both sides of an inequality by a coefficient, make sure to check if the coefficient is positive or negative to determine the direction of the inequality sign.
When solving inequalities, it's essential to check the solution to ensure that it satisfies the original inequality.

Final Thoughts

Solving inequalities is a crucial concept in mathematics that helps us understand the relationship between different variables. By following the steps outlined in this article, we can solve the inequality $5(a+16) \ \textgreater \ 20$ to find the value of $a$ . This type of problem is commonly encountered in algebra and is an essential skill to master for students and professionals alike.

Introduction

In our previous article, we solved the inequality $5(a+16) \ \textgreater \ 20$ to find the value of $a$ . In this article, we will answer some frequently asked questions related to solving inequalities.

Q&A

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

A: The first step in solving an inequality is to simplify the inequality by distributing coefficients to terms inside parentheses, combining like terms, and isolating the variable on one side of the inequality.

Q: How do I know which direction to change the inequality sign when subtracting or adding terms to both sides?

A: When subtracting or adding terms to both sides of an inequality, you need to change the direction of the inequality sign if the term being subtracted or added is negative. If the term being subtracted or added is positive, you do not need to change the direction of the inequality sign.

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

A: Solving an inequality is similar to solving an equation, but with one key difference: when solving an inequality, you need to consider two possible solutions, one for each direction of the inequality sign. When solving an equation, you only need to find one solution.

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 consider both parts of the compound inequality when solving.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, plug the solution back into the original inequality and see if it satisfies the inequality. If it does, then your solution is correct.

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

A: Some common mistakes to avoid when solving inequalities include forgetting to distribute coefficients to terms inside parentheses, forgetting to change the direction of the inequality sign when subtracting or adding terms to both sides, and forgetting to check the solution to ensure that it satisfies the original inequality.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, you need to be careful to enter the correct values and to check the solution to ensure that it satisfies the original inequality.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that the correct steps are taken.
When distributing coefficients to terms inside parentheses, make sure to multiply the coefficient to each term separately.
When subtracting or adding terms to both sides of an inequality, make sure to change the direction of the inequality sign if necessary.
When dividing both sides of an inequality by a coefficient, make sure to check if the coefficient is positive or negative to determine the direction of the inequality sign.

Real-World Applications

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid making mistakes such as forgetting to distribute coefficients to terms inside parentheses or forgetting to change the direction of the inequality sign when subtracting or adding terms to both sides.
When dividing both sides of an inequality by a coefficient, make sure to check if the coefficient is positive or negative to determine the direction of the inequality sign.
When solving inequalities, it's essential to check the solution to ensure that it satisfies the original inequality.