Solve For \[$d\$\].$\[ \begin{array}{c} \frac{d}{4} - 5 \ \textless \ 6 \\ d \ \textless \ [?] \end{array} \\]

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than or less than symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving a specific inequality involving the variable $d$. We will break down the steps to solve the inequality and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $\frac{d}{4} - 5 < 6$. Our goal is to solve for $d$ and find the value that satisfies the inequality.

Step 1: Add 5 to Both Sides

To isolate the term involving $d$, we need to get rid of the constant term on the left side of the inequality. We can do this by adding 5 to both sides of the inequality.

$\frac{d}{4} - 5 + 5 < 6 + 5$

This simplifies to:

$\frac{d}{4} < 11$

Step 2: Multiply Both Sides by 4

To get rid of the fraction, we can multiply both sides of the inequality by 4.

$\frac{d}{4} \times 4 < 11 \times 4$

This simplifies to:

$d < 44$

Conclusion

We have successfully solved the inequality and found the value of $d$. The final answer is $d < 44$. This means that any value of $d$ that is less than 44 will satisfy the original inequality.

Why is Solving Inequalities Important?

Solving inequalities is an essential skill in mathematics, as it allows us to compare values and make decisions based on those comparisons. Inequalities are used in a wide range of applications, from finance to science, to model real-world problems and make predictions.

Real-World Applications of Inequalities

Inequalities are used in many real-world applications, including:

• Finance: Inequalities are used to model investment returns, interest rates, and other financial metrics.
• Science: Inequalities are used to model physical phenomena, such as the motion of objects and the behavior of chemical reactions.
• Engineering: Inequalities are used to design and optimize systems, such as bridges and buildings.

Tips and Tricks for Solving Inequalities

Here are some tips and tricks for solving inequalities:

• Use inverse operations: To isolate the variable, use inverse operations, such as addition and subtraction, multiplication and division.
• Get rid of fractions: To get rid of fractions, multiply both sides of the inequality by the denominator.
• Check your work: Always check your work by plugging in values to ensure that the inequality is satisfied.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving inequalities:

• Not checking your work: Failing to check your work can lead to incorrect solutions.
• Not using inverse operations: Failing to use inverse operations can make it difficult to isolate the variable.
• Not getting rid of fractions: Failing to get rid of fractions can make the inequality difficult to solve.

Conclusion

Introduction

In our previous article, we discussed the steps involved in solving inequalities. In this article, we will answer some frequently asked questions about solving inequalities. Whether you are a student, a teacher, or simply someone who wants to improve their math skills, this article is for you.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than or less than symbols. For example, $x > 5$ is an inequality that compares the value of $x$ to 5.

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. This involves using inverse operations, such as addition and subtraction, multiplication and division, to get rid of any constants or fractions that are attached to the variable.

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

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to first factor the quadratic expression, if possible. Then, you need to use the sign of the quadratic expression to determine the intervals where the inequality is satisfied.

Q: What is the significance of the inequality sign?

A: The inequality sign is used to indicate the direction of the inequality. For example, the inequality sign $<$ means that the value of the variable is less than the value on the other side of the inequality sign.

Q: Can I use the same steps to solve all types of inequalities?

A: No, the steps to solve different types of inequalities may vary. For example, to solve a quadratic inequality, you need to use a different set of steps than to solve a linear inequality.

Q: How do I check my work when solving an inequality?

A: To check your work, you need to plug in values from the solution set into the original inequality to ensure that it is satisfied.

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

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

• Not checking your work
• Not using inverse operations
• Not getting rid of fractions
• Not considering the direction of the inequality sign

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities, but you need to be careful to enter the correct values and to check your work.

Conclusion

Solving inequalities is an essential skill in mathematics, and it requires a clear understanding of the steps involved. By following the steps outlined in this article and avoiding common mistakes, you can solve inequalities with confidence and accuracy. Remember to check your work and to consider the direction of the inequality sign to ensure that you are solving the inequality correctly.