Solve The Inequality For $u$.$u - 9 \ \textless \ -7$Simplify Your Answer As Much As Possible. $\square$

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Introduction

Inequalities are mathematical expressions that compare two values, often with a greater-than or less-than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $u - 9 < -7$ and simplify our answer as much as possible.

Understanding the Inequality

The given inequality is $u - 9 < -7$. To solve this inequality, we need to isolate the variable $u$. The first step is to add $9$ to both sides of the inequality. This will help us get rid of the negative term on the left-hand side.

Adding 9 to Both Sides

When we add $9$ to both sides of the inequality, we get:

$$u - 9 + 9 < -7 + 9$$

This simplifies to:

$$u < 2$$

Simplifying the Inequality

Now that we have isolated the variable $u$, we can simplify the inequality. The inequality $u < 2$ means that $u$ is less than $2$. This can be written as:

$$u \in (-\infty, 2)$$

Interpreting the Solution

The solution to the inequality $u - 9 < -7$ is $u \in (-\infty, 2)$. This means that any value of $u$ that is less than $2$ is a solution to the inequality. In other words, the set of all possible values of $u$ that satisfy the inequality is the interval $(-\infty, 2)$.

Graphical Representation

To visualize the solution to the inequality, we can graph the interval $(-\infty, 2)$ on a number line. The number line represents all possible values of $u$, and the interval $(-\infty, 2)$ represents the set of all values of $u$ that satisfy the inequality.

Conclusion

Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we solved the inequality $u - 9 < -7$ and simplified our answer as much as possible. We added $9$ to both sides of the inequality to isolate the variable $u$, and then simplified the inequality to get $u < 2$. The solution to the inequality is $u \in (-\infty, 2)$, which represents the set of all possible values of $u$ that satisfy the inequality.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes. Here are a few:

• Not adding the same value to both sides: When adding a value to both sides of an inequality, make sure to add the same value to both sides. For example, if we add $3$ to both sides of the inequality $u - 9 < -7$, we get $u - 6 < -4$, not $u - 9 < -4$.
• Not simplifying the inequality: After adding a value to both sides of an inequality, make sure to simplify the inequality. For example, if we add $9$ to both sides of the inequality $u - 9 < -7$, we get $u < 2$, not $u < 2 + 9$.
• Not considering the direction of the inequality: When solving an inequality, make sure to consider the direction of the inequality. For example, if we have the inequality $u - 9 > -7$, we need to add $9$ to both sides to get $u > 2$, not $u < 2$.

Real-World Applications

Solving inequalities has many real-world applications. Here are a few:

• Finance: In finance, inequalities are used to model financial situations. For example, if we have a savings account with a balance of $u$ dollars, and we want to know how much money we have after $9$ years, we can use the inequality $u - 9 < -7$ to model the situation.
• Science: In science, inequalities are used to model scientific situations. For example, if we have a population of $u$ individuals, and we want to know how many individuals will be left after $9$ years, we can use the inequality $u - 9 < -7$ to model the situation.
• Engineering: In engineering, inequalities are used to model engineering situations. For example, if we have a bridge with a length of $u$ meters, and we want to know how much weight the bridge can hold, we can use the inequality $u - 9 < -7$ to model the situation.

Conclusion

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Introduction

In our previous article, we discussed how to solve inequalities and provided a step-by-step guide to finding the value of $u$ in the inequality $u - 9 < -7$. In this article, we will answer some frequently asked questions about solving inequalities.

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

A: An inequality is a mathematical expression that compares two values, often with a greater-than or less-than symbol. An equation, on the other hand, is a mathematical expression that states that two values are equal. For example, the inequality $u - 9 < -7$ is different from the equation $u - 9 = -7$.

Q: How do I know which direction to add when solving an inequality?

A: When solving an inequality, you need to add the same value to both sides of the inequality. If the inequality is of the form $u - a < b$, you need to add $a$ to both sides to isolate the variable $u$. If the inequality is of the form $u - a > b$, you need to add $a$ to both sides to isolate the variable $u$.

Q: Can I multiply or divide both sides of an inequality?

A: Yes, you can multiply or divide both sides of an inequality, but you need to be careful. When multiplying or dividing both sides of an inequality, you need to make sure that you are not changing the direction of the inequality. For example, if you have the inequality $u - 9 < -7$ and you multiply both sides by $-1$, you get $-u + 9 > 7$, which is a different inequality.

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

A: To determine if an inequality is true or false, you need to plug in a value for the variable $u$ and see if the inequality holds true. For example, if you have the inequality $u - 9 < -7$ and you plug in $u = 2$, you get $2 - 9 < -7$, which is true.

Q: Can I use inequalities to solve systems of equations?

A: Yes, you can use inequalities to solve systems of equations. Inequalities can be used to model real-world situations, and solving systems of inequalities can help you find the solution to a system of equations.

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 line that represents the boundary of the inequality. If the inequality is of the form $u - a < b$, you need to draw a line that is open on the left side and closed on the right side. If the inequality is of the form $u - a > b$, you need to draw a line that is closed on the left side and open on the right side.

Q: Can I use inequalities to model real-world situations?

A: Yes, you can use inequalities to model real-world situations. Inequalities can be used to model situations where there are constraints or limitations. For example, if you have a budget of $u$ dollars and you want to know how much money you can spend on a particular item, you can use an inequality to model the situation.

Conclusion

Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we answered some frequently asked questions about solving inequalities and provided a step-by-step guide to finding the solution to an inequality. We also discussed how to graph an inequality on a number line and how to use inequalities to model real-world situations.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes. Here are a few:

• Not adding the same value to both sides: When adding a value to both sides of an inequality, make sure to add the same value to both sides. For example, if we add $3$ to both sides of the inequality $u - 9 < -7$, we get $u - 6 < -4$, not $u - 9 < -4$.
• Not simplifying the inequality: After adding a value to both sides of an inequality, make sure to simplify the inequality. For example, if we add $9$ to both sides of the inequality $u - 9 < -7$, we get $u < 2$, not $u < 2 + 9$.
• Not considering the direction of the inequality: When solving an inequality, make sure to consider the direction of the inequality. For example, if we have the inequality $u - 9 > -7$, we need to add $9$ to both sides to get $u > 2$, not $u < 2$.

Real-World Applications

Solving inequalities has many real-world applications. Here are a few:

• Finance: In finance, inequalities are used to model financial situations. For example, if we have a savings account with a balance of $u$ dollars, and we want to know how much money we have after $9$ years, we can use the inequality $u - 9 < -7$ to model the situation.
• Science: In science, inequalities are used to model scientific situations. For example, if we have a population of $u$ individuals, and we want to know how many individuals will be left after $9$ years, we can use the inequality $u - 9 < -7$ to model the situation.
• Engineering: In engineering, inequalities are used to model engineering situations. For example, if we have a bridge with a length of $u$ meters, and we want to know how much weight the bridge can hold, we can use the inequality $u - 9 < -7$ to model the situation.