Solve The Inequality:$\frac{1}{3}c - 2 \frac{1}{2} \geq \frac{1}{2}$

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, while keeping the inequality sign intact. In this article, we will focus on solving the inequality $\frac{1}{3}c - 2 \frac{1}{2} \geq \frac{1}{2}$.

Understanding the Inequality

The given inequality is $\frac{1}{3}c - 2 \frac{1}{2} \geq \frac{1}{2}$. To solve this inequality, we need to isolate the variable $c$ on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality by combining the fractions.

Simplifying the Left-Hand Side

To simplify the left-hand side, we need to find a common denominator for the fractions. The least common multiple of 3 and 2 is 6. We can rewrite the fractions with a common denominator of 6:

$$\frac{1}{3}c = \frac{2}{6}c$$

$$-2 \frac{1}{2} = -\frac{5}{2} = -\frac{15}{6}$$

Now, we can rewrite the left-hand side of the inequality as:

$$\frac{2}{6}c - \frac{15}{6} \geq \frac{1}{2}$$

Combining the Fractions

To combine the fractions, we need to find a common denominator. In this case, the common denominator is 6. We can rewrite the fractions with a common denominator of 6:

$$\frac{2}{6}c - \frac{15}{6} = \frac{2c - 15}{6}$$

Now, we can rewrite the inequality as:

$$\frac{2c - 15}{6} \geq \frac{1}{2}$$

Multiplying Both Sides by 6

To eliminate the fraction, we can multiply both sides of the inequality by 6:

$$2c - 15 \geq 3$$

Adding 15 to Both Sides

To isolate the term with the variable, we can add 15 to both sides of the inequality:

$$2c \geq 18$$

Dividing Both Sides by 2

To solve for the variable, we can divide both sides of the inequality by 2:

$$c \geq 9$$

Conclusion

In this article, we solved the inequality $\frac{1}{3}c - 2 \frac{1}{2} \geq \frac{1}{2}$ by simplifying the left-hand side, combining the fractions, multiplying both sides by 6, adding 15 to both sides, and dividing both sides by 2. The solution to the inequality is $c \geq 9$.

Tips and Tricks

• When solving inequalities, it's essential to keep the inequality sign intact.
• When combining fractions, find a common denominator and rewrite the fractions with that denominator.
• When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.
• When adding or subtracting both sides of an inequality by a number, keep the inequality sign the same.

Real-World Applications

Solving inequalities has numerous real-world applications, including:

• Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
• Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
• Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes

• Flipping the Inequality Sign: When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.
• Not Finding a Common Denominator: When combining fractions, find a common denominator and rewrite the fractions with that denominator.
• Not Keeping the Inequality Sign: When solving inequalities, keep the inequality sign intact.

Conclusion

Introduction

In our previous article, we solved the inequality $\frac{1}{3}c - 2 \frac{1}{2} \geq \frac{1}{2}$ by simplifying the left-hand side, combining the fractions, multiplying both sides by 6, adding 15 to both sides, and dividing both sides by 2. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

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

A: An equation is a statement that two expressions are equal, while an inequality is a statement that one expression is greater than or less than another expression.

Q: How do I know which direction to flip 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, flip the inequality sign. For example, if you have the inequality $x > 3$ and you multiply both sides by -1, the inequality becomes $x < -3$.

Q: What is the least common multiple (LCM) and how do I find it?

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. To find the LCM, list the multiples of each number and find the smallest number that appears in all the lists.

Q: How do I simplify an inequality with fractions?

A: To simplify an inequality with fractions, find a common denominator and rewrite the fractions with that denominator. Then, combine the fractions by adding or subtracting the numerators.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, first factor the quadratic expression, if possible. Then, use the sign chart method to determine the intervals where the inequality is true.

Q: What is the sign chart method and how do I use it?

A: The sign chart method is a technique used to determine the intervals where a quadratic inequality is true. To use the sign chart method, first factor the quadratic expression, if possible. Then, create a sign chart by listing the intervals where the quadratic expression is positive or negative.

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

A: To graph an inequality on a number line, first find the solution set of the inequality. Then, plot the solution set on a number line by using a closed circle for a greater than or equal to inequality and an open circle for a less than or equal to inequality.

Conclusion

Solving inequalities is a crucial skill in mathematics, with numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to keep the inequality sign intact, find a common denominator when combining fractions, and flip the inequality sign when multiplying or dividing both sides by a negative number. With practice and patience, you can master the art of solving inequalities.

Tips and Tricks

• When solving inequalities, it's essential to keep the inequality sign intact.
• When combining fractions, find a common denominator and rewrite the fractions with that denominator.
• When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.
• When adding or subtracting both sides of an inequality by a number, keep the inequality sign the same.

Real-World Applications

Solving inequalities has numerous real-world applications, including:

• Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
• Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
• Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes

• Flipping the Inequality Sign: When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.
• Not Finding a Common Denominator: When combining fractions, find a common denominator and rewrite the fractions with that denominator.
• Not Keeping the Inequality Sign: When solving inequalities, keep the inequality sign intact.

Conclusion

Solving inequalities is a crucial skill in mathematics, with numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to keep the inequality sign intact, find a common denominator when combining fractions, and flip the inequality sign when multiplying or dividing both sides by a negative number. With practice and patience, you can master the art of solving inequalities.