Solve The Inequality For W W W : 8 7 W − 1 ≥ 5 7 W − 2 3 \frac{8}{7}w - 1 \geq \frac{5}{7}w - \frac{2}{3} 7 8 ​ W − 1 ≥ 7 5 ​ W − 3 2 ​ Simplify Your Answer As Much As Possible.

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than or less than symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $\frac{8}{7}w - 1 \geq \frac{5}{7}w - \frac{2}{3}$ and simplify our answer as much as possible.

Understanding the Inequality

The given inequality is $\frac{8}{7}w - 1 \geq \frac{5}{7}w - \frac{2}{3}$. To solve this inequality, we need to isolate the variable $w$ on one side of the inequality sign. We can start by adding $1$ to both sides of the inequality to get rid of the negative term.

Adding 1 to Both Sides

$\frac{8}{7}w - 1 + 1 \geq \frac{5}{7}w - \frac{2}{3} + 1$

This simplifies to:

$\frac{8}{7}w \geq \frac{5}{7}w - \frac{2}{3} + 1$

Combining Like Terms

To combine like terms, we need to find a common denominator for the fractions. The least common multiple of $7$ and $3$ is $21$. We can rewrite the inequality as:

$\frac{8}{7}w \geq \frac{15}{21}w - \frac{14}{21} + \frac{21}{21}$

This simplifies to:

$\frac{8}{7}w \geq \frac{15}{21}w + \frac{7}{21}$

Subtracting $\frac{15}{21}w$ from Both Sides

To isolate the variable $w$, we need to subtract $\frac{15}{21}w$ from both sides of the inequality.

$\frac{8}{7}w - \frac{15}{21}w \geq \frac{15}{21}w + \frac{7}{21} - \frac{15}{21}w$

This simplifies to:

$\frac{24}{21}w \geq \frac{7}{21}$

Dividing Both Sides by $\frac{24}{21}$

To solve for $w$, we need to divide both sides of the inequality by $\frac{24}{21}$.

$\frac{\frac{24}{21}w}{\frac{24}{21}} \geq \frac{\frac{7}{21}}{\frac{24}{21}}$

This simplifies to:

$w \geq \frac{7}{24}$

Conclusion

In conclusion, the solution to the inequality $\frac{8}{7}w - 1 \geq \frac{5}{7}w - \frac{2}{3}$ is $w \geq \frac{7}{24}$. This means that the value of $w$ must be greater than or equal to $\frac{7}{24}$ to satisfy the inequality.

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 variables such as supply and demand. In finance, inequalities can be used to model the relationship between interest rates and investment returns. In engineering, inequalities can be used to model the relationship between physical quantities such as velocity and acceleration.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations (PEMDAS) and to simplify the inequality as much as possible. Additionally, it's crucial to check the solution by plugging it back into the original inequality to ensure that it satisfies the inequality.

Common Mistakes

When solving inequalities, it's common to make mistakes such as:

• Not following the order of operations (PEMDAS)
• Not simplifying the inequality as much as possible
• Not checking the solution by plugging it back into the original inequality

Conclusion

Solving inequalities is a crucial skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to always follow the order of operations (PEMDAS) and to simplify the inequality as much as possible. Additionally, it's essential to check the solution by plugging it back into the original inequality to ensure that it satisfies the inequality.