Solve The Inequality:$\[ \frac{a+2}{16}-\frac{a+3}{12}\ \textgreater \ \frac{a+6}{24} \\]Choose The Correct Answer:A. \[$a \ \textgreater \ -6\$\] B. \[$a \ \textless \ -6\$\] C. \[$a \ \textgreater \ 6\$\] D.

Introduction

Inequalities are mathematical expressions that compare two values, often with a greater than or less than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving a specific inequality and provide a step-by-step guide on how to approach it.

The Inequality

The given inequality is:

$$\frac{a+2}{16}-\frac{a+3}{12}\ \textgreater \ \frac{a+6}{24}$$

Our goal is to solve for $a$ and determine the correct answer among the options provided.

Step 1: Simplify the Inequality

To simplify the inequality, we need to find a common denominator for the fractions. The least common multiple (LCM) of 16, 12, and 24 is 48.

\frac{a+2}{16}-\frac{a+3}{12} = \frac{(a+2)(3)}{48}-\frac{(a+3)(4)}{48}

Now, we can combine the fractions:

\frac{(a+2)(3)-(a+3)(4)}{48} = \frac{3a+6-4a-12}{48}

Simplifying further, we get:

\frac{-a-6}{48} = \frac{-a}{48}-\frac{6}{48}

Now, we can rewrite the inequality as:

$$\frac{-a}{48}-\frac{6}{48}\ \textgreater \ \frac{a+6}{24}$$

Step 2: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the inequality by 48:

(-a-6)\  \textgreater \  2(a+6)

Expanding the right-hand side, we get:

-a-6\  \textgreater \  2a+12

Step 3: Isolate the Variable

To isolate the variable $a$, we need to get all the terms with $a$ on one side of the inequality. We can do this by adding 6 to both sides:

-a\  \textgreater \  2a+18

Next, we can add $a$ to both sides to get:

0\  \textgreater \  3a+18

Step 4: Solve for $a$

To solve for $a$, we need to isolate the variable on one side of the inequality. We can do this by subtracting 18 from both sides:

-18\  \textgreater \  3a

Next, we can divide both sides by 3 to get:

-a\  \textgreater \  -6

Conclusion

The final answer is $a\ \textless \ -6$. Therefore, the correct answer is:

B. $a\ \textless \ -6$

Discussion

Solving inequalities can be a challenging task, but with a step-by-step approach, it can be made easier. In this article, we solved a specific inequality and provided a guide on how to approach it. We simplified the inequality, eliminated the fractions, isolated the variable, and solved for $a$. The final answer is $a\ \textless \ -6$, which is the correct answer among the options provided.

Tips and Tricks

• When solving inequalities, it's essential to follow the order of operations (PEMDAS).
• Simplifying the inequality can make it easier to solve.
• Eliminating fractions can make the inequality more manageable.
• Isolating the variable can help you find the solution.
• Solving for $a$ can be done by isolating the variable on one side of the inequality.

Practice Problems

Solving inequalities can be a challenging task, but with practice, you can become more confident. Here are some practice problems to help you improve your skills:

1. Solve the inequality: $\frac{x+2}{3}-\frac{x+1}{4}\ \textgreater \ \frac{x+5}{12}$
2. Solve the inequality: $\frac{y-2}{5}+\frac{y+1}{3}\ \textless \ \frac{y+4}{15}$
3. Solve the inequality: $\frac{z+1}{2}-\frac{z-2}{3}\ \textgreater \ \frac{z+3}{6}$

Introduction

In our previous article, we solved a specific inequality and provided a step-by-step guide on how to approach it. In this article, we will answer some frequently asked questions (FAQs) about solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often with a greater than or less than symbol.

Q: How do I simplify an inequality?

A: To simplify an inequality, you need to find a common denominator for the fractions. The least common multiple (LCM) of the denominators is the common denominator.

Q: How do I eliminate fractions in an inequality?

A: To eliminate fractions in an inequality, you can multiply both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable in an inequality, you need to get all the terms with the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality.

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, you need to find the roots of the quadratic equation $ax^2 + bx + c = 0$. The roots of the equation are the values of $x$ that make the equation true. You can use the quadratic formula to find the roots of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to find the roots of a quadratic equation. The formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to solve a quadratic inequality?

A: To use the quadratic formula to solve a quadratic inequality, you need to find the roots of the quadratic equation $ax^2 + bx + c = 0$. You can then use the roots to determine the intervals where the inequality is true.

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

A: Some common mistakes to avoid when solving inequalities include:

• Not simplifying the inequality before solving it
• Not eliminating fractions before solving the inequality
• Not isolating the variable before solving the inequality
• Not checking the solution to make sure it is true

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become more confident. Remember to simplify the inequality, eliminate fractions, isolate the variable, and check the solution to make sure it is true. If you have any more questions or need further clarification, please don't hesitate to ask.

Practice Problems

Here are some practice problems to help you improve your skills:

1. Solve the inequality: $\frac{x+2}{3}-\frac{x+1}{4}\ \textgreater \ \frac{x+5}{12}$
2. Solve the inequality: $\frac{y-2}{5}+\frac{y+1}{3}\ \textless \ \frac{y+4}{15}$
3. Solve the inequality: $\frac{z+1}{2}-\frac{z-2}{3}\ \textgreater \ \frac{z+3}{6}$

I hope this article has helped you understand how to solve inequalities. Remember to practice regularly to improve your skills.