Solve The Compound Inequality: − 4 ( 6 X + 1 ) ≤ 8 X − 36 ≤ 4 -4(6x + 1) \leq 8x - 36 \leq 4 − 4 ( 6 X + 1 ) ≤ 8 X − 36 ≤ 4

$$

Introduction

In mathematics, inequalities are used to compare the values of different expressions. A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will focus on solving a compound inequality of the form $-4(6x + 1) \leq 8x - 36 \leq 4$. This type of inequality involves multiple expressions and requires careful manipulation to isolate the variable.

Understanding the Compound Inequality

The given compound inequality is $-4(6x + 1) \leq 8x - 36 \leq 4$. To solve this inequality, we need to break it down into two separate inequalities: $-4(6x + 1) \leq 8x - 36$ and $8x - 36 \leq 4$. We will then solve each inequality separately and combine the solutions to find the final answer.

Solving the First Inequality

The first inequality is $-4(6x + 1) \leq 8x - 36$. To solve this inequality, we need to expand the left-hand side and simplify the expression.

# Expand the left-hand side of the inequality
import sympy as sp
x = sp.symbols('x')
inequality = -4*(6*x + 1)
expanded_inequality = sp.expand(inequality)
print(expanded_inequality)

The expanded inequality is $-24x - 4 \leq 8x - 36$. Now, we can add $24x$ to both sides of the inequality to get $-4 \leq 32x - 36$.

Solving the Second Inequality

The second inequality is $8x - 36 \leq 4$. To solve this inequality, we need to add $36$ to both sides of the inequality to get $8x \leq 40$.

Combining the Solutions

Now that we have solved both inequalities, we can combine the solutions to find the final answer. The first inequality is $-4 \leq 32x - 36$, and the second inequality is $8x \leq 40$. We can add $36$ to both sides of the first inequality to get $32 \leq 32x$, and then divide both sides by $32$ to get $1 \leq x$.

Similarly, we can divide both sides of the second inequality by $8$ to get $x \leq 5$.

Finding the Final Answer

The final answer is the intersection of the two solutions: $1 \leq x \leq 5$. This means that the solution to the compound inequality is all values of $x$ that are greater than or equal to $1$ and less than or equal to $5$.

Conclusion

In this article, we solved a compound inequality of the form $-4(6x + 1) \leq 8x - 36 \leq 4$. We broke down the inequality into two separate inequalities and solved each one separately. We then combined the solutions to find the final answer, which is $1 \leq x \leq 5$. This type of problem requires careful manipulation of the expressions and attention to detail to ensure that the solution is correct.

Step-by-Step Solution

Here is a step-by-step solution to the compound inequality:

1. Expand the left-hand side of the first inequality: $-4(6x + 1) \leq 8x - 36$
2. Simplify the expression: $-24x - 4 \leq 8x - 36$
3. Add $24x$ to both sides of the inequality: $-4 \leq 32x - 36$
4. Add $36$ to both sides of the inequality: $32 \leq 32x$
5. Divide both sides of the inequality by $32$: $1 \leq x$
6. Solve the second inequality: $8x - 36 \leq 4$
7. Add $36$ to both sides of the inequality: $8x \leq 40$
8. Divide both sides of the inequality by $8$: $x \leq 5$
9. Combine the solutions: $1 \leq x \leq 5$

Frequently Asked Questions

• What is a compound inequality? A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".
• How do I solve a compound inequality? To solve a compound inequality, you need to break it down into two or more separate inequalities and solve each one separately. You can then combine the solutions to find the final answer.
• What is the solution to the compound inequality $-4(6x + 1) \leq 8x - 36 \leq 4$? The solution to the compound inequality is $1 \leq x \leq 5$.
  

Introduction

In our previous article, we solved a compound inequality of the form $-4(6x + 1) \leq 8x - 36 \leq 4$. We broke down the inequality into two separate inequalities and solved each one separately. In this article, we will answer some frequently asked questions about compound inequalities.

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to break it down into two or more separate inequalities and solve each one separately. You can then combine the solutions to find the final answer.

Q: What is the difference between a compound inequality and a single inequality?

A: A single inequality is a statement that compares two expressions, such as $x > 2$. A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".

Q: How do I know which logical operator to use when combining inequalities?

A: When combining inequalities, you need to use the logical operator that makes sense in the context of the problem. For example, if you are looking for values of $x$ that satisfy both inequalities, you would use the "and" operator.

Q: Can I use a compound inequality to solve a system of equations?

A: Yes, you can use a compound inequality to solve a system of equations. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

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. You can use a Venn diagram or a number line to help you visualize the solution.

Q: Can I use a compound inequality to solve a problem that involves multiple variables?

A: Yes, you can use a compound inequality to solve a problem that involves multiple variables. By combining the inequalities, you can find the values of the variables that satisfy all of the conditions.

Q: How do I know if a compound inequality is true or false?

A: To determine if a compound inequality is true or false, you need to evaluate the inequality for each value of the variable. If the inequality is true for all values of the variable, then it is true. If the inequality is false for any value of the variable, then it is false.

Q: Can I use a compound inequality to solve a problem that involves absolute values?

A: Yes, you can use a compound inequality to solve a problem that involves absolute values. By combining the inequalities, you can find the values of the variable that satisfy all of the conditions.

Q: How do I simplify a compound inequality?

A: To simplify a compound inequality, you need to combine like terms and eliminate any unnecessary variables. You can also use algebraic manipulations to simplify the inequality.

Q: Can I use a compound inequality to solve a problem that involves fractions?

A: Yes, you can use a compound inequality to solve a problem that involves fractions. By combining the inequalities, you can find the values of the variable that satisfy all of the conditions.

Q: How do I know if a compound inequality is linear or nonlinear?

A: To determine if a compound inequality is linear or nonlinear, you need to examine the inequality and determine if it can be written in the form $ax + b \leq c$ or $ax^2 + bx + c \leq 0$.

Q: Can I use a compound inequality to solve a problem that involves quadratic equations?

A: Yes, you can use a compound inequality to solve a problem that involves quadratic equations. By combining the inequalities, you can find the values of the variable that satisfy all of the conditions.

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

A: To graph a compound inequality on a number line, you need to graph each inequality separately and then combine the graphs. You can use a Venn diagram or a number line to help you visualize the solution.

Q: Can I use a compound inequality to solve a problem that involves systems of linear equations?

A: Yes, you can use a compound inequality to solve a problem that involves systems of linear equations. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I know if a compound inequality is consistent or inconsistent?

A: To determine if a compound inequality is consistent or inconsistent, you need to examine the inequality and determine if it has a solution. If the inequality has a solution, then it is consistent. If the inequality does not have a solution, then it is inconsistent.

Q: Can I use a compound inequality to solve a problem that involves matrices?

A: Yes, you can use a compound inequality to solve a problem that involves matrices. By combining the inequalities, you can find the values of the variables that satisfy all of the conditions.

Q: How do I simplify a compound inequality with absolute values?

A: To simplify a compound inequality with absolute values, you need to combine like terms and eliminate any unnecessary variables. You can also use algebraic manipulations to simplify the inequality.

Q: Can I use a compound inequality to solve a problem that involves systems of nonlinear equations?

A: Yes, you can use a compound inequality to solve a problem that involves systems of nonlinear equations. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I graph a compound inequality with absolute values?

A: To graph a compound inequality with absolute values, you need to graph each inequality separately and then combine the graphs. You can use a Venn diagram or a number line to help you visualize the solution.

Q: Can I use a compound inequality to solve a problem that involves systems of equations with multiple variables?

A: Yes, you can use a compound inequality to solve a problem that involves systems of equations with multiple variables. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I know if a compound inequality is true or false when it involves absolute values?

A: To determine if a compound inequality is true or false when it involves absolute values, you need to evaluate the inequality for each value of the variable. If the inequality is true for all values of the variable, then it is true. If the inequality is false for any value of the variable, then it is false.

Q: Can I use a compound inequality to solve a problem that involves systems of equations with fractions?

A: Yes, you can use a compound inequality to solve a problem that involves systems of equations with fractions. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I simplify a compound inequality with fractions?

A: To simplify a compound inequality with fractions, you need to combine like terms and eliminate any unnecessary variables. You can also use algebraic manipulations to simplify the inequality.

Q: Can I use a compound inequality to solve a problem that involves systems of equations with quadratic expressions?

A: Yes, you can use a compound inequality to solve a problem that involves systems of equations with quadratic expressions. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I graph a compound inequality with fractions?

A: To graph a compound inequality with fractions, you need to graph each inequality separately and then combine the graphs. You can use a Venn diagram or a number line to help you visualize the solution.

Q: Can I use a compound inequality to solve a problem that involves systems of equations with absolute values and fractions?

A: Yes, you can use a compound inequality to solve a problem that involves systems of equations with absolute values and fractions. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I simplify a compound inequality with absolute values and fractions?

A: To simplify a compound inequality with absolute values and fractions, you need to combine like terms and eliminate any unnecessary variables. You can also use algebraic manipulations to simplify the inequality.

Q: Can I use a compound inequality to solve a problem that involves systems of equations with multiple variables, absolute values, and fractions?

A: Yes, you can use a compound inequality to solve a problem that involves systems of equations with multiple variables, absolute values, and fractions. By combining the inequalities, you can find the values of the variables that satisfy all of the equations.

Q: How do I graph a compound inequality with multiple variables, absolute values, and fractions?

A: To graph a compound inequality with multiple variables, absolute values, and fractions, you need to graph each inequality separately and then combine the graphs. You can use a Venn diagram or a number line to help you visualize the solution.

Conclusion

In this article, we have answered some frequently asked questions about compound inequalities. We have discussed how to solve compound inequalities, how to graph them, and how to simplify them. We have also discussed how to use compound inequalities to solve problems that involve multiple variables, absolute values, and fractions. By following these steps, you can use compound inequalities to solve a wide range of problems in mathematics.