Solve This Inequality: $\frac{j}{4} - 8 \ \textless \ 4$.A) $j \ \textless \ -48$ B) $j \ \textless \ 12$ C) $j \ \textless \ 48$ D) $j \ \textgreater \ -12$

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 maintaining the direction of the inequality. In this article, we will focus on solving the inequality $\frac{j}{4} - 8 \ \textless \ 4$.

Understanding the Inequality

The given inequality is $\frac{j}{4} - 8 \ \textless \ 4$. To solve this inequality, we need to isolate the variable $j$ on one side of the inequality sign. The first step is to add $8$ to both sides of the inequality, which will eliminate the constant term on the left-hand side.

Step 1: Add 8 to Both Sides

$\frac{j}{4} - 8 + 8 \ \textless \ 4 + 8$

This simplifies to:

$\frac{j}{4} \ \textless \ 12$

Step 2: Multiply Both Sides by 4

To isolate the variable $j$, we need to multiply both sides of the inequality by $4$. This will eliminate the fraction on the left-hand side.

$4 \times \frac{j}{4} \ \textless \ 4 \times 12$

This simplifies to:

$j \ \textless \ 48$

Conclusion

Therefore, the solution to the inequality $\frac{j}{4} - 8 \ \textless \ 4$ is $j \ \textless \ 48$. This means that any value of $j$ that is less than $48$ will satisfy the given inequality.

Comparison with Answer Choices

Let's compare our solution with the answer choices:

• A) $j \ \textless \ -48$: This is not correct, as our solution is $j \ \textless \ 48$.
• B) $j \ \textless \ 12$: This is not correct, as our solution is $j \ \textless \ 48$.
• C) $j \ \textless \ 48$: This is correct, as our solution is $j \ \textless \ 48$.
• D) $j \ \textgreater \ -12$: This is not correct, as our solution is $j \ \textless \ 48$.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always add or subtract the same value to both sides of the inequality.
• Multiply or divide both sides of the inequality by the same non-zero value.
• Be careful when multiplying or dividing both sides of the inequality by a negative value, as it will change the direction of the inequality.
• Use inverse operations to isolate the variable on one side of the inequality.

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving inequalities:

1. Solve the inequality $2x + 5 \ \textless \ 11$.
2. Solve the inequality $x - 3 \ \textgreater \ 7$.
3. Solve the inequality $\frac{x}{2} + 2 \ \textless \ 9$.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities by isolating the variable on one side of the inequality sign. In this article, we will provide a Q&A guide to help you reinforce your understanding of solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign, while maintaining the direction of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides by the same non-zero value.

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 $ax + b \ \textless \ c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c \ \textless \ 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the sign of the quadratic expression to determine the solution set. You can also use the quadratic formula to find the roots of the quadratic equation, and then use the sign of the quadratic expression to determine the solution set.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict inequality symbol, such as $<$ or $>$, while a non-strict inequality is an inequality that uses a non-strict inequality symbol, such as $\leq$ or $\geq$.

Q: How do I determine the solution set of an inequality?

A: To determine the solution set of an inequality, you need to isolate the variable on one side of the inequality sign, while maintaining the direction of the inequality. You can then use the sign of the inequality to determine the solution set.

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

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

• Adding or subtracting the wrong value to both sides of the inequality
• Multiplying or dividing both sides of the inequality by the wrong value
• Not maintaining the direction of the inequality
• Not checking the solution set for extraneous solutions

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems, such as those found in this article. You can also try solving inequalities on your own, using a calculator or computer program to check your work.

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to add or subtract the same value to both sides of the inequality, multiply or divide both sides by the same non-zero value, and use inverse operations to isolate the variable. With practice, you will become proficient in solving inequalities and be able to tackle more complex problems.

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving inequalities:

1. Solve the inequality $2x + 5 \ \textless \ 11$.
2. Solve the inequality $x - 3 \ \textgreater \ 7$.
3. Solve the inequality $\frac{x}{2} + 2 \ \textless \ 9$.

Answer Key

1. $x \ \textless \ 3$
2. $x \ \textgreater \ 10$
3. $x \ \textless \ 14$

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always add or subtract the same value to both sides of the inequality.
• Multiply or divide both sides of the inequality by the same non-zero value.
• Be careful when multiplying or dividing both sides of the inequality by a negative value, as it will change the direction of the inequality.
• Use inverse operations to isolate the variable on one side of the inequality.
• Check the solution set for extraneous solutions.