Solve For $n$. $-5 \geq \frac{n}{4}$ Write The Solution As An Inequality (for Example, $n\ \textgreater \ 9$). $\square$

by ADMIN 131 views

Understanding the Inequality

When solving for an unknown variable in an inequality, we need to isolate the variable on one side of the inequality sign. In this case, we are given the inequality −5≥n4-5 \geq \frac{n}{4}, and we need to solve for the variable nn. To do this, we will use the properties of inequalities and algebraic operations to isolate the variable.

Isolating the Variable

To isolate the variable nn, we need to get rid of the fraction on the right-hand side of the inequality. We can do this by multiplying both sides of the inequality by the denominator, which is 4. This will give us:

−5≥n4-5 \geq \frac{n}{4}

−5×4≥n4×4-5 \times 4 \geq \frac{n}{4} \times 4

−20≥n-20 \geq n

Writing the Solution as an Inequality

Now that we have isolated the variable nn, we can write the solution as an inequality. Since the inequality is greater than or equal to, we will use the ≥\geq symbol to indicate this. Therefore, the solution to the inequality is:

n≤−20n \leq -20

Checking the Solution

To check our solution, we can plug in a value of nn that satisfies the inequality and see if it is true. Let's try plugging in n=−21n = -21:

−5≥−214-5 \geq \frac{-21}{4}

−5≥−5.25-5 \geq -5.25

Since −5-5 is indeed greater than or equal to −5.25-5.25, our solution is correct.

Conclusion

In this article, we solved the inequality −5≥n4-5 \geq \frac{n}{4} and wrote the solution as an inequality. We used the properties of inequalities and algebraic operations to isolate the variable nn and then wrote the solution in the correct format. We also checked our solution by plugging in a value of nn that satisfies the inequality.

Tips and Tricks

  • When solving inequalities, always remember to check your solution by plugging in a value of the variable that satisfies the inequality.
  • Use the properties of inequalities to isolate the variable on one side of the inequality sign.
  • Be careful when multiplying or dividing both sides of an inequality by a negative number, as this can change the direction of the inequality.

Common Mistakes

  • Failing to check the solution by plugging in a value of the variable that satisfies the inequality.
  • Not using the properties of inequalities to isolate the variable on one side of the inequality sign.
  • Making mistakes when multiplying or dividing both sides of an inequality by a negative number.

Real-World Applications

  • Inequalities are used in many real-world applications, such as finance, economics, and engineering.
  • In finance, inequalities are used to model the behavior of financial instruments, such as stocks and bonds.
  • In economics, inequalities are used to model the behavior of economic systems, such as supply and demand.
  • In engineering, inequalities are used to model the behavior of physical systems, such as electrical circuits and mechanical systems.

Final Thoughts

Solving inequalities is an important skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve inequalities and write the solution in the correct format. Remember to check your solution by plugging in a value of the variable that satisfies the inequality, and be careful when multiplying or dividing both sides of an inequality by a negative number. With practice and patience, you can become proficient in solving inequalities and apply this skill to many real-world problems.

Frequently Asked Questions

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

A: The first step in solving an inequality is to isolate the variable on one side of the inequality sign. This can be done by using the properties of inequalities, such as adding or subtracting the same value to both sides, or multiplying or dividing both sides by the same non-zero value.

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

A: When multiplying or dividing both sides of an inequality by a negative number, you need to change the direction of the inequality sign. For example, if you have the inequality a≥ba \geq b and you multiply both sides by −1-1, the inequality becomes −a≤−b-a \leq -b.

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 the inequality a≥ba \geq b and you add 55 to both sides, the inequality becomes a+5≥b+5a + 5 \geq b + 5.

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 plug in a value of the variable that satisfies the inequality and see if it is true. For example, if you have the inequality a≥ba \geq b and you plug in a=5a = 5 and b=3b = 3, the inequality is true because 5≥35 \geq 3.

Q: Can I multiply or divide both sides of an inequality by a fraction?

A: Yes, you can multiply or divide both sides of an inequality by a fraction. However, you need to be careful when doing this, as it can change the direction of the inequality sign. For example, if you have the inequality a≥ba \geq b and you multiply both sides by 12\frac{1}{2}, the inequality becomes 12a≥12b\frac{1}{2}a \geq \frac{1}{2}b.

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

A: To solve an inequality with a variable on both sides, you need to isolate the variable on one side of the inequality sign. This can be done by using the properties of inequalities, such as adding or subtracting the same value to both sides, or multiplying or dividing both sides by the same non-zero value.

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

A: Yes, you can use the same steps to solve a compound inequality. A compound inequality is an inequality that contains two or more inequalities joined by the word "and" or "or". For example, the compound inequality a≥ba \geq b and c≤dc \leq d can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to an inequality is valid?

A: To determine if a solution to an inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

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

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator, as it can give you an incorrect solution if you are not careful.

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 sign.

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

A: Yes, you can use the same steps to solve a linear inequality. A linear inequality is an inequality that contains a linear expression on one side of the inequality sign. For example, the linear inequality a≥ba \geq b can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a linear inequality is valid?

A: To determine if a solution to a linear inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

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

A: Yes, you can use the same steps to solve a quadratic inequality. A quadratic inequality is an inequality that contains a quadratic expression on one side of the inequality sign. For example, the quadratic inequality a2≥b2a^2 \geq b^2 can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a quadratic inequality is valid?

A: To determine if a solution to a quadratic inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

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

A: Yes, you can use the same steps to solve a rational inequality. A rational inequality is an inequality that contains a rational expression on one side of the inequality sign. For example, the rational inequality ab≥c\frac{a}{b} \geq c can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a rational inequality is valid?

A: To determine if a solution to a rational inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

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

A: Yes, you can use the same steps to solve a polynomial inequality. A polynomial inequality is an inequality that contains a polynomial expression on one side of the inequality sign. For example, the polynomial inequality a3≥b3a^3 \geq b^3 can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a polynomial inequality is valid?

A: To determine if a solution to a polynomial inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

Q: Can I use the same steps to solve a trigonometric inequality?

A: Yes, you can use the same steps to solve a trigonometric inequality. A trigonometric inequality is an inequality that contains a trigonometric expression on one side of the inequality sign. For example, the trigonometric inequality sin⁡x≥cos⁡x\sin x \geq \cos x can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a trigonometric inequality is valid?

A: To determine if a solution to a trigonometric inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

Q: Can I use the same steps to solve a logarithmic inequality?

A: Yes, you can use the same steps to solve a logarithmic inequality. A logarithmic inequality is an inequality that contains a logarithmic expression on one side of the inequality sign. For example, the logarithmic inequality log⁡x≥log⁡y\log x \geq \log y can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a logarithmic inequality is valid?

A: To determine if a solution to a logarithmic inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

Q: Can I use the same steps to solve an exponential inequality?

A: Yes, you can use the same steps to solve an exponential inequality. An exponential inequality is an inequality that contains an exponential expression on one side of the inequality sign. For example, the exponential inequality ax≥bxa^x \geq b^x can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to an exponential inequality is valid?

A: To determine if a solution to an exponential inequality is valid, you need to check if the solution satisfies the inequality. This can be done by plugging in the solution into the inequality and seeing if it is true.

Q: Can I use the same steps to solve a system of inequalities?

A: Yes, you can use the same steps to solve a system of inequalities. A system of inequalities is a set of two or more inequalities that are related to each other. For example, the system of inequalities a≥ba \geq b and c≤dc \leq d can be solved by isolating the variable on one side of the inequality sign.

Q: How do I know if a solution to a system of inequalities is valid?

A: To determine if a solution to a system of inequalities is valid, you need to check if the solution satisfies each of the inequalities in the system. This can be done by plugging in the solution into each of the inequalities and seeing if it is true.

Q: Can I use the same steps to solve a linear programming problem?

A: Yes, you can use the same steps to solve a linear programming problem. A linear programming problem is a problem that involves maximizing or minimizing a