Solve The Inequality:${ 5t \ \textgreater \ C = \left(\frac{5}{2}\right)^{18} }$

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while keeping the inequality sign intact. In this article, we will focus on solving the inequality $5t > c = \left(\frac{5}{2}\right)^{18}$.

Understanding the Inequality

The given inequality is $5t > c = \left(\frac{5}{2}\right)^{18}$. To solve this inequality, we need to isolate the variable $t$ on one side of the inequality sign. The first step is to simplify the right-hand side of the inequality by evaluating the expression $\left(\frac{5}{2}\right)^{18}$.

Evaluating the Right-Hand Side

To evaluate the expression $\left(\frac{5}{2}\right)^{18}$, we can use the property of exponents that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. Applying this property, we get:

$$\left(\frac{5}{2}\right)^{18} = \frac{5^{18}}{2^{18}}$$

Now, we can evaluate the expression $5^{18}$ and $2^{18}$ separately.

Evaluating $5^{18}$ and $2^{18}$

To evaluate $5^{18}$ and $2^{18}$, we can use a calculator or a computer program. Alternatively, we can use the property of exponents that states $a^n = (a^m)^{\frac{n}{m}}$. Applying this property, we get:

$$5^{18} = (5^6)^3 = (15625)^3 = 30517578125$$

$$2^{18} = (2^9)^2 = (512)^2 = 262144$$

Now, we can substitute these values back into the expression $\frac{5^{18}}{2^{18}}$.

Substituting the Values

Substituting the values of $5^{18}$ and $2^{18}$ back into the expression $\frac{5^{18}}{2^{18}}$, we get:

$$\frac{5^{18}}{2^{18}} = \frac{30517578125}{262144} = 115.5$$

Now, we can rewrite the original inequality as $5t > 115.5$.

Solving the Inequality

To solve the inequality $5t > 115.5$, we can divide both sides of the inequality by 5.

Dividing Both Sides

Dividing both sides of the inequality $5t > 115.5$ by 5, we get:

$$t > \frac{115.5}{5}$$

Evaluating the Right-Hand Side

To evaluate the right-hand side of the inequality $t > \frac{115.5}{5}$, we can divide 115.5 by 5.

Dividing 115.5 by 5

Dividing 115.5 by 5, we get:

$$\frac{115.5}{5} = 23.1$$

Now, we can rewrite the inequality as $t > 23.1$.

Conclusion

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In conclusion, to solve the inequality $5t > c = \left(\frac{5}{2}\right)^{18}$, we need to isolate the variable $t$ on one side of the inequality sign. We can do this by simplifying the right-hand side of the inequality, evaluating the expression $\left(\frac{5}{2}\right)^{18}$, and then dividing both sides of the inequality by 5. The final solution to the inequality is $t > 23.1$.

Real-World Applications

Solving inequalities has many real-world applications in fields such as economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between two variables, such as the price of a good and the quantity demanded. In finance, inequalities can be used to model the relationship between the interest rate and the amount of money borrowed. In engineering, inequalities can be used to model the relationship between the speed of a vehicle and the distance traveled.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. One common mistake is to forget to isolate the variable on one side of the inequality sign. Another common mistake is to forget to keep the inequality sign intact when dividing both sides of the inequality by a number. To avoid these mistakes, it is essential to carefully read and understand the problem, and to follow the steps outlined above.

Tips and Tricks

When solving inequalities, there are several tips and tricks that can help. One tip is to use a calculator or a computer program to evaluate expressions and simplify fractions. Another tip is to use the property of exponents that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to simplify expressions. Additionally, it is essential to carefully read and understand the problem, and to follow the steps outlined above.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities and provided a step-by-step guide on how to isolate the variable on one side of the inequality sign. In this article, we will answer some frequently asked questions about solving inequalities and provide additional tips and tricks to help you become more confident in your ability to solve inequalities.

Q&A

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

A: An inequality is a mathematical expression that compares two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. An equation, on the other hand, is a mathematical expression that states that two values are equal.

Q: How do I know which direction to move the inequality sign when I multiply or divide both sides of the inequality by a negative number?

A: When you multiply or divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality $x > 5$ and you multiply both sides by -1, the inequality becomes $x < -5$.

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 keep the inequality signs intact when solving the compound inequality.

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 substitute a value for the variable and see if the inequality holds true. For example, if you have the inequality $x > 5$ and you substitute $x = 6$, the inequality becomes $6 > 5$, which is true.

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 operations into the calculator.

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

A: An inequality is linear if it can be written in the form $ax > b$ or $ax < b$, where $a$ and $b$ are constants. An inequality is non-linear if it cannot be written in this form.

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

A: No, you cannot use the same steps to solve a quadratic inequality as you would to solve a linear inequality. Quadratic inequalities require a different set of steps to solve.

Q: How do I know if an inequality is a strict inequality or a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x > 5$. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $x ≥ 5$.

Tips and Tricks

Tip 1: Use a calculator to evaluate expressions and simplify fractions.

Using a calculator can save you time and effort when solving inequalities.

Tip 2: Use the property of exponents that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to simplify expressions.

This property can help you simplify complex expressions and make solving inequalities easier.

Tip 3: Be careful when multiplying or dividing both sides of the inequality by a negative number.

Reversing the direction of the inequality sign is crucial when multiplying or dividing both sides of the inequality by a negative number.

Tip 4: Use a graphing calculator to visualize the solution to an inequality.

Graphing calculators can help you visualize the solution to an inequality and make it easier to understand.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics that has many real-world applications. By following the steps outlined in this article and using the tips and tricks provided, you can become more confident in your ability to solve inequalities. Remember to be careful when multiplying or dividing both sides of the inequality by a negative number, and to use a calculator to evaluate expressions and simplify fractions.