Solve The Inequality:$\[ 1 + \frac{x}{3} \ \textless \ -\frac{9}{5} \\]

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than, less than, or equal to symbols. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $1 + \frac{x}{3} < -\frac{9}{5}$.

Understanding the Inequality

The given inequality is $1 + \frac{x}{3} < -\frac{9}{5}$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the right-hand side of the inequality by finding a common denominator.

Simplifying the Right-Hand Side

To simplify the right-hand side, we need to find a common denominator for the fractions $1$ and $-\frac{9}{5}$. The least common multiple of $1$ and $5$ is $5$, so we can rewrite the inequality as:

$$1 + \frac{x}{3} < -\frac{9}{5}$$

$$\frac{5}{5} + \frac{x}{3} < -\frac{9}{5}$$

$$\frac{5}{5} + \frac{15x}{15} < -\frac{9}{5}$$

$$\frac{15x}{15} + \frac{5}{5} < -\frac{9}{5}$$

$$\frac{15x + 5}{15} < -\frac{9}{5}$$

Subtracting 1 from Both Sides

Now that we have simplified the right-hand side, we can subtract $1$ from both sides of the inequality to isolate the term with the variable $x$.

$$\frac{15x + 5}{15} - 1 < -\frac{9}{5} - 1$$

$$\frac{15x + 5}{15} - \frac{15}{15} < -\frac{9}{5} - \frac{5}{5}$$

$$\frac{15x + 5 - 15}{15} < -\frac{14}{5}$$

$$\frac{15x - 10}{15} < -\frac{14}{5}$$

Multiplying Both Sides by 15

To eliminate the fraction, we can multiply both sides of the inequality by $15$.

$$\frac{15x - 10}{15} \cdot 15 < -\frac{14}{5} \cdot 15$$

$$15x - 10 < -42$$

Adding 10 to Both Sides

Now that we have eliminated the fraction, we can add $10$ to both sides of the inequality to isolate the term with the variable $x$.

$$15x - 10 + 10 < -42 + 10$$

$$15x < -32$$

Dividing Both Sides by 15

Finally, we can divide both sides of the inequality by $15$ to solve for the variable $x$.

$$\frac{15x}{15} < \frac{-32}{15}$$

$$x < -\frac{32}{15}$$

Conclusion

In this article, we have solved the inequality $1 + \frac{x}{3} < -\frac{9}{5}$. We simplified the right-hand side, subtracted $1$ from both sides, multiplied both sides by $15$, added $10$ to both sides, and finally divided both sides by $15$ to solve for the variable $x$. The solution to the inequality is $x < -\frac{32}{15}$.

Example Use Case

Suppose we want to find the values of $x$ that satisfy the inequality $1 + \frac{x}{3} < -\frac{9}{5}$. We can use the solution to the inequality, $x < -\frac{32}{15}$, to find the values of $x$ that satisfy the inequality. For example, if we want to find the values of $x$ that satisfy the inequality when $x$ is between $-2$ and $-1$, we can substitute $x = -2$ and $x = -1$ into the inequality and check if the inequality is true.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations and to simplify the right-hand side of the inequality before isolating the variable. Additionally, when multiplying or dividing both sides of the inequality by a negative number, we need to reverse the direction of the inequality sign.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Some common mistakes include:

• Not simplifying the right-hand side of the inequality before isolating the variable
• Not following the order of operations
• Not reversing the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number

Conclusion

Introduction

In our previous article, we solved the inequality $1 + \frac{x}{3} < -\frac{9}{5}$. In this article, we will answer some common questions about solving inequalities.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to simplify the right-hand side of the inequality by finding a common denominator.

Q: How do I simplify the right-hand side of an inequality?

A: To simplify the right-hand side of an inequality, you need to find a common denominator for the fractions. You can do this by multiplying the numerator and denominator of each fraction by the same number.

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

A: The order of operations when solving an inequality is the same as when solving an equation:

1. Simplify the right-hand side of the inequality
2. Subtract or add the same value to both sides of the inequality
3. Multiply or divide both sides of the inequality by the same value
4. Check the direction of the inequality sign

Q: How do I know when to reverse the direction of the inequality sign?

A: You need to reverse the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number.

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

A: The main difference between solving an inequality and solving an equation is that when solving an inequality, you need to consider all possible values of the variable, whereas when solving an equation, you need to find a single value of the variable.

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

A: No, you cannot use the same methods to solve all types of inequalities. For example, when solving a linear inequality, you can use the methods outlined in this article. However, when solving a quadratic inequality, you may need to use different methods.

Q: How do I know when to use a different method to solve an inequality?

A: You need to use a different method to solve an inequality when the inequality is quadratic or when the inequality involves absolute values.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator to solve an inequality, as the calculator may not always give you the correct answer.

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

A: To check your answer when solving an inequality, you need to substitute the value of the variable into the original inequality and check if the inequality is true.

Conclusion

In conclusion, solving inequalities involves finding the values of the variable that make the inequality true. By following the steps outlined in this article and using the Q&A guide, you can solve inequalities and find the values of the variable that satisfy the inequality.

Example Use Case

Suppose we want to find the values of $x$ that satisfy the inequality $1 + \frac{x}{3} < -\frac{9}{5}$. We can use the solution to the inequality, $x < -\frac{32}{15}$, to find the values of $x$ that satisfy the inequality. For example, if we want to find the values of $x$ that satisfy the inequality when $x$ is between $-2$ and $-1$, we can substitute $x = -2$ and $x = -1$ into the inequality and check if the inequality is true.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations and to simplify the right-hand side of the inequality before isolating the variable. Additionally, when multiplying or dividing both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Some common mistakes include:

• Not simplifying the right-hand side of the inequality before isolating the variable
• Not following the order of operations
• Not reversing the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number

Conclusion

In conclusion, solving inequalities involves finding the values of the variable that make the inequality true. By following the steps outlined in this article and using the Q&A guide, you can solve inequalities and find the values of the variable that satisfy the inequality. Remember to simplify the right-hand side of the inequality, follow the order of operations, and reverse the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number.