Solve The Inequality:$\[ 5x - 7 \ \textgreater \ 9 \\]

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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, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality $5x - 7 > 9$ using step-by-step instructions and explanations.

Understanding the Inequality

The given inequality is $5x - 7 > 9$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to add $7$ to both sides of the inequality, which will eliminate the negative term on the left-hand side.

Adding 7 to Both Sides

When we add $7$ to both sides of the inequality, we get:

$$5x - 7 + 7 > 9 + 7$$

This simplifies to:

$$5x > 16$$

Dividing Both Sides by 5

To isolate the variable $x$, we need to divide both sides of the inequality by $5$. However, when we divide both sides of an inequality by a negative number, the direction of the inequality sign changes. In this case, we are dividing both sides by a positive number, so the direction of the inequality sign remains the same.

$$\frac{5x}{5} > \frac{16}{5}$$

This simplifies to:

$$x > \frac{16}{5}$$

Conclusion

In conclusion, the solution to the inequality $5x - 7 > 9$ is $x > \frac{16}{5}$. This means that any value of $x$ greater than $\frac{16}{5}$ will satisfy the inequality.

Graphical Representation

To visualize the solution to the inequality, we can graph the inequality on a number line. The number line represents all possible values of $x$, and the inequality $x > \frac{16}{5}$ indicates that all values of $x$ greater than $\frac{16}{5}$ are solutions to the inequality.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between supply and demand, while in finance, inequalities can be used to calculate interest rates and investment returns. In engineering, inequalities can be used to design and optimize systems, such as electrical circuits and mechanical systems.

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.
• When dividing both sides of an inequality by a negative number, the direction of the inequality sign changes.
• Use a number line to visualize the solution to the inequality.
• Check your solution by plugging in a value of $x$ that satisfies the inequality.

Common Mistakes

When solving inequalities, it's common to make mistakes, such as:

• Adding or subtracting the wrong value to both sides of the inequality.
• Dividing both sides of the inequality by a negative number without changing the direction of the inequality sign.
• Failing to check the solution by plugging in a value of $x$ that satisfies the inequality.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the step-by-step instructions and explanations in this article, you can learn how to solve inequalities and apply this skill to various fields. Remember to always add or subtract the same value to both sides of the inequality, use a number line to visualize the solution, and check your solution by plugging in a value of $x$ that satisfies the inequality.

Final Answer

The final answer to the inequality $5x - 7 > 9$ is $x > \frac{16}{5}$.

Introduction

In our previous article, we discussed how to solve the inequality $5x - 7 > 9$ using step-by-step instructions and explanations. In this article, we will provide a Q&A guide to help you better understand how to solve inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than or less than 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 by multiplying or dividing both sides of the inequality 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 $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can add or subtract the same value to both sides of the inequality, or multiply or divide both sides of the inequality by the same non-zero value. For example, to solve the inequality $2x + 3 > 5$, you can subtract $3$ from both sides of the inequality to get $2x > 2$, and then divide both sides of the inequality by $2$ to get $x > 1$.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can factor the quadratic expression on the left-hand side of the inequality, or use the quadratic formula to find the solutions to the quadratic equation. For example, to solve the inequality $x^2 + 4x + 4 > 0$, you can factor the quadratic expression to get $(x + 2)^2 > 0$, and then use the fact that the square of a real number is always non-negative to conclude that the inequality is true for all real values of $x$.

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

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x > 2$ or $x < 3$. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $x \geq 2$ or $x \leq 3$.

Q: How do I solve a strict inequality?

A: To solve a strict inequality, you can add or subtract the same value to both sides of the inequality, or multiply or divide both sides of the inequality by the same non-zero value. For example, to solve the inequality $x > 2$, you can add $1$ to both sides of the inequality to get $x + 1 > 3$, and then subtract $1$ from both sides of the inequality to get $x > 2$.

Q: How do I solve a non-strict inequality?

A: To solve a non-strict inequality, you can add or subtract the same value to both sides of the inequality, or multiply or divide both sides of the inequality by the same non-zero value. For example, to solve the inequality $x \geq 2$, you can add $1$ to both sides of the inequality to get $x + 1 \geq 3$, and then subtract $1$ from both sides of the inequality to get $x \geq 2$.

Q: What is the importance of solving inequalities?

A: Solving inequalities is an essential skill in mathematics that has numerous real-world applications. Inequalities are used to model real-world problems, such as the relationship between supply and demand in economics, the growth of a population in biology, and the design of electrical circuits in engineering.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through examples and exercises in a textbook or online resource. You can also try solving inequalities on your own, using a calculator or computer program to check your answers.

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 a zero value.
• Failing to check the solution by plugging in a value of $x$ that satisfies the inequality.
• Failing to consider the direction of the inequality sign.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the step-by-step instructions and explanations in this article, you can learn how to solve inequalities and apply this skill to various fields. Remember to always add or subtract the same value to both sides of the inequality, use a calculator or computer program to check your answers, and consider the direction of the inequality sign.