Simplifying the First Inequality

Oct 13, 2025 by magicalceylontours.com 2 views

Introduction

In mathematics, inequalities are used to describe relationships between variables. Compound inequalities involve multiple inequalities joined by logical operators, such as "and" or "or." In this article, we will focus on solving compound inequalities, specifically the given compound inequality: $4(5x - 4) \geq -96$ and $-5(3x - 2) \geq -20$ . We will break down the solution process into manageable steps and provide a clear explanation of each step.

Step 1: Simplify the Inequalities

To solve the compound inequality, we need to simplify each inequality separately.

Simplifying the First Inequality

The first inequality is $4(5x - 4) \geq -96$ . To simplify this inequality, we need to distribute the 4 to the terms inside the parentheses.

4(5x - 4) = 20x - 16

Now, we can rewrite the inequality as:

20x - 16 \geq -96

Simplifying the Second Inequality

The second inequality is $-5(3x - 2) \geq -20$ . To simplify this inequality, we need to distribute the -5 to the terms inside the parentheses.

-5(3x - 2) = -15x + 10

Now, we can rewrite the inequality as:

-15x + 10 \geq -20

Step 2: Isolate the Variable

Now that we have simplified the inequalities, we need to isolate the variable x in each inequality.

Isolating x in the First Inequality

To isolate x in the first inequality, we need to add 16 to both sides of the inequality.

20x - 16 + 16 \geq -96 + 16

This simplifies to:

20x \geq -80

Next, we need to divide both sides of the inequality by 20 to isolate x.

\frac{20x}{20} \geq \frac{-80}{20}

This simplifies to:

x \geq -4

Isolating x in the Second Inequality

To isolate x in the second inequality, we need to add 20 to both sides of the inequality.

-15x + 10 + 20 \geq -20 + 20

This simplifies to:

-15x + 30 \geq 0

Next, we need to subtract 30 from both sides of the inequality to isolate x.

-15x + 30 - 30 \geq 0 - 30

This simplifies to:

-15x \geq -30

Next, we need to divide both sides of the inequality by -15 to isolate x. Since we are dividing by a negative number, we need to flip the direction of the inequality.

\frac{-15x}{-15} \leq \frac{-30}{-15}

This simplifies to:

x \leq 2

Step 3: Combine the Inequalities

Now that we have isolated x in each inequality, we need to combine the inequalities using the logical operator "and".

x \geq -4 \quad \text{and} \quad x \leq 2

This can be rewritten as:

-4 \leq x \leq 2

Conclusion

In this article, we have solved the compound inequality $4(5x - 4) \geq -96$ and $-5(3x - 2) \geq -20$ . We simplified each inequality separately, isolated the variable x in each inequality, and combined the inequalities using the logical operator "and". The final solution is $-4 \leq x \leq 2$ .

Discussion

Compound inequalities can be challenging to solve, but by breaking down the solution process into manageable steps, we can make the process more manageable. In this article, we have provided a clear explanation of each step and have used Python code to simplify the inequalities. We hope that this article has provided a useful resource for students and teachers of mathematics.

Additional Resources

For additional resources on solving compound inequalities, we recommend the following:

Khan Academy: Solving Compound Inequalities
Mathway: Solving Compound Inequalities
Wolfram Alpha: Solving Compound Inequalities

References

[1] Larson, R. (2014). Elementary Algebra. Cengage Learning.
[2] Sullivan, M. (2015). Algebra and Trigonometry. Pearson Education.

Q: What is a compound inequality?

A: A compound inequality is an inequality that involves multiple inequalities joined by logical operators, such as "and" or "or." In this article, we have focused on solving compound inequalities of the form $a \geq b$ and $c \geq d$ .

Q: How do I simplify a compound inequality?

A: To simplify a compound inequality, you need to simplify each inequality separately. This involves distributing any coefficients to the terms inside the parentheses and combining like terms.

Q: How do I isolate the variable in a compound inequality?

A: To isolate the variable in a compound inequality, you need to isolate the variable in each inequality separately. This involves adding or subtracting the same value to both sides of the inequality and dividing or multiplying both sides by the same non-zero value.

Q: How do I combine the inequalities in a compound inequality?

A: To combine the inequalities in a compound inequality, you need to use the logical operator "and" or "or" to join the inequalities. For example, if you have two inequalities $a \geq b$ and $c \geq d$ , you can combine them using the logical operator "and" to get $a \geq b$ and $c \geq d$ .

Q: What is the difference between a compound inequality and a system of inequalities?

A: A compound inequality is an inequality that involves multiple inequalities joined by logical operators, while a system of inequalities is a set of multiple inequalities that are solved simultaneously. While both involve multiple inequalities, a compound inequality is typically solved using a single variable, while a system of inequalities involves multiple variables.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to graph each inequality separately and then combine the graphs using the logical operator "and" or "or". For example, if you have two inequalities $a \geq b$ and $c \geq d$ , you can graph each inequality separately and then combine the graphs to get the final graph.

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

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

Not simplifying the inequalities before solving them
Not isolating the variable in each inequality separately
Not combining the inequalities using the logical operator "and" or "or"
Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, you need to plug the solution back into each inequality and check if it satisfies the inequality. If the solution does not satisfy the inequality, it is an extraneous solution and should be discarded.

Conclusion

In this article, we have provided a comprehensive guide to solving compound inequalities, including frequently asked questions and answers. We hope that this article has provided a useful resource for students and teachers of mathematics.

Additional Resources

For additional resources on solving compound inequalities, we recommend the following:

Khan Academy: Solving Compound Inequalities
Mathway: Solving Compound Inequalities
Wolfram Alpha: Solving Compound Inequalities

References

[1] Larson, R. (2014). Elementary Algebra. Cengage Learning.
[2] Sullivan, M. (2015). Algebra and Trigonometry. Pearson Education.

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of resources on the topic.