Solve For $h$.$6(h-18)-11 \leq 1$

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Introduction

In mathematics, solving inequalities is a crucial aspect of algebraic expressions. It involves isolating the variable of interest, in this case, $h$, to determine its possible values. The given inequality is $6(h-18)-11 \leq 1$, and our goal is to solve for $h$. This involves simplifying the expression, isolating the variable, and determining the range of values that satisfy the inequality.

Understanding the Inequality

The given inequality is $6(h-18)-11 \leq 1$. To begin solving, we need to simplify the expression by evaluating the terms inside the parentheses. We can start by distributing the 6 to the terms inside the parentheses: $6(h-18) = 6h - 108$. Now, the inequality becomes $6h - 108 - 11 \leq 1$.

Simplifying the Inequality

Next, we need to combine like terms to simplify the inequality. We can combine the constants -108 and -11 to get -119. The inequality now becomes $6h - 119 \leq 1$. To isolate the variable $h$, we need to get rid of the constant term -119. We can do this by adding 119 to both sides of the inequality: $6h - 119 + 119 \leq 1 + 119$.

Isolating the Variable

After adding 119 to both sides, the inequality becomes $6h \leq 120$. Now, we need to isolate the variable $h$ by dividing both sides of the inequality by 6. However, we need to be careful when dividing by a negative number, as it will change the direction of the inequality. Since 6 is positive, the inequality remains the same: $h \leq 20$.

Conclusion

In conclusion, we have solved the inequality $6(h-18)-11 \leq 1$ for $h$. By simplifying the expression, isolating the variable, and determining the range of values that satisfy the inequality, we found that $h \leq 20$. This means that any value of $h$ less than or equal to 20 will satisfy the given inequality.

Graphical Representation

To visualize the solution, we can graph the inequality on a number line. The inequality $h \leq 20$ can be represented as a closed circle at 20, indicating that $h$ can be equal to 20. The inequality also includes all values less than 20, which can be represented as an open circle at 20. The number line will show that $h$ can take on any value less than or equal to 20.

Real-World Applications

Solving inequalities like $6(h-18)-11 \leq 1$ has numerous real-world applications. In finance, inequalities are used to determine the maximum or minimum value of an investment. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to determine the effectiveness of a treatment. By solving inequalities, we can make informed decisions and optimize our choices.

Common Mistakes

When solving inequalities, there are several common mistakes to avoid. One mistake is to forget to change the direction of the inequality when dividing by a negative number. Another mistake is to not simplify the expression before isolating the variable. By being aware of these common mistakes, we can avoid errors and ensure that our solutions are accurate.

Tips and Tricks

To solve inequalities like $6(h-18)-11 \leq 1$, there are several tips and tricks to keep in mind. One tip is to simplify the expression as much as possible before isolating the variable. Another tip is to use inverse operations to isolate the variable. By using these tips and tricks, we can solve inequalities efficiently and accurately.

Final Thoughts

In conclusion, solving inequalities like $6(h-18)-11 \leq 1$ requires careful attention to detail and a thorough understanding of algebraic expressions. By simplifying the expression, isolating the variable, and determining the range of values that satisfy the inequality, we can solve for $h$. This has numerous real-world applications and can be used to make informed decisions and optimize our choices. By following the tips and tricks outlined in this article, we can solve inequalities efficiently and accurately.

Frequently Asked Questions

Q: What is the solution to the inequality $6(h-18)-11 \leq 1$?

A: The solution to the inequality $6(h-18)-11 \leq 1$ is $h \leq 20$.

Q: How do I simplify the expression before isolating the variable?

A: To simplify the expression, combine like terms and evaluate the terms inside the parentheses.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle represents a value that is included in the solution, while an open circle represents a value that is not included in the solution.

Q: How do I avoid common mistakes when solving inequalities?

A: To avoid common mistakes, be aware of the direction of the inequality when dividing by a negative number and simplify the expression before isolating the variable.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has numerous real-world applications, including finance, engineering, and medicine.

Q: How do I use inverse operations to isolate the variable?

A: To use inverse operations, apply the opposite operation to both sides of the inequality. For example, if the inequality is $h + 3 \leq 5$, subtract 3 from both sides to isolate the variable.

Introduction

Solving inequalities can be a challenging task, but with the right guidance, it can be made easier. In this article, we will provide a comprehensive Q&A section to help you understand and solve inequalities. Whether you are a student, teacher, or simply someone who wants to learn more about inequalities, this article is for you.

Q&A Section

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two expressions using a symbol such as <, >, ≤, or ≥.

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

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

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing inverse operations. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is the order of operations when solving an inequality?

A: The order of operations is the same as when solving an equation: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: How do I handle negative numbers when solving an inequality?

A: When dividing or multiplying both sides of an inequality by a negative number, you need to change the direction of the inequality.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle represents a value that is included in the solution, while an open circle represents a value that is not included in the solution.

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

A: To graph an inequality on a number line, you need to draw a closed circle at the endpoint of the solution and an open circle at the endpoint that is not included in the solution.

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

A: Some common mistakes to avoid when solving inequalities include forgetting to change the direction of the inequality when dividing by a negative number, not simplifying the expression before isolating the variable, and not using inverse operations to isolate the variable.

Q: How do I use inverse operations to isolate the variable?

A: To use inverse operations, apply the opposite operation to both sides of the inequality. For example, if the inequality is $h + 3 \leq 5$, subtract 3 from both sides to isolate the variable.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has numerous real-world applications, including finance, engineering, and medicine.

Q: How do I use inequalities to make informed decisions?

A: Inequalities can be used to make informed decisions by comparing different options and determining which one is the best choice.

Q: What are some tips and tricks for solving inequalities?

A: Some tips and tricks for solving inequalities include simplifying the expression before isolating the variable, using inverse operations to isolate the variable, and being aware of the direction of the inequality when dividing by a negative number.

Conclusion

Solving inequalities can be a challenging task, but with the right guidance, it can be made easier. By following the tips and tricks outlined in this article, you can solve inequalities efficiently and accurately. Whether you are a student, teacher, or simply someone who wants to learn more about inequalities, this article is for you.

Frequently Asked Questions

Q: What is the solution to the inequality $6(h-18)-11 \leq 1$?

A: The solution to the inequality $6(h-18)-11 \leq 1$ is $h \leq 20$.

Q: How do I simplify the expression before isolating the variable?

A: To simplify the expression, combine like terms and evaluate the terms inside the parentheses.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle represents a value that is included in the solution, while an open circle represents a value that is not included in the solution.

Q: How do I avoid common mistakes when solving inequalities?

A: To avoid common mistakes, be aware of the direction of the inequality when dividing by a negative number and simplify the expression before isolating the variable.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has numerous real-world applications, including finance, engineering, and medicine.

Q: How do I use inverse operations to isolate the variable?

A: To use inverse operations, apply the opposite operation to both sides of the inequality. For example, if the inequality is $h + 3 \leq 5$, subtract 3 from both sides to isolate the variable.

Additional Resources

• Inequality Solver
• Inequality Grapher
• Inequality Calculator

Conclusion

Solving inequalities can be a challenging task, but with the right guidance, it can be made easier. By following the tips and tricks outlined in this article, you can solve inequalities efficiently and accurately. Whether you are a student, teacher, or simply someone who wants to learn more about inequalities, this article is for you.