Solve The Inequality Algebraically: ${ \frac{x+11}{x-18} \leq 1 }$The Solution Is { \square$}$(Simplify Your Answer. Type Your Answer In Interval Notation.)

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Introduction

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In algebra, inequalities are mathematical statements that compare two expressions and indicate whether one is greater than, less than, or equal to the other. Solving inequalities algebraically involves manipulating the inequality to isolate the variable and find the solution set. In this article, we will focus on solving the inequality $\frac{x+11}{x-18} \leq 1$ algebraically and provide a step-by-step guide on how to do it.

Understanding the Inequality

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The given inequality is $\frac{x+11}{x-18} \leq 1$. To solve this inequality, we need to isolate the variable $x$ and find the solution set. The first step is to get rid of the fraction by multiplying both sides of the inequality by the denominator, which is $x-18$. However, we need to be careful when multiplying by a negative number, as it will change the direction of the inequality.

Multiplying by a Negative Number

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When we multiply both sides of the inequality by $x-18$, we need to consider two cases: when $x-18$ is positive and when $x-18$ is negative. If $x-18$ is positive, then the inequality will remain the same. However, if $x-18$ is negative, then the inequality will change direction.

Case 1: $x-18 > 0$

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When $x-18 > 0$, we can multiply both sides of the inequality by $x-18$ without changing the direction of the inequality. This gives us:

$$\frac{x+11}{x-18} \leq 1 \Rightarrow x+11 \leq x-18$$

Simplifying the inequality, we get:

$$11 \leq -18$$

This is a contradiction, as $11$ is not less than or equal to $-18$. Therefore, there are no solutions to the inequality when $x-18 > 0$.

Case 2: $x-18 < 0$

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When $x-18 < 0$, we need to change the direction of the inequality when multiplying by $x-18$. This gives us:

$$\frac{x+11}{x-18} \leq 1 \Rightarrow x+11 \geq x-18$$

Simplifying the inequality, we get:

$$11 \geq -18$$

This is a true statement, as $11$ is indeed greater than or equal to $-18$. Therefore, the solution to the inequality when $x-18 < 0$ is all real numbers $x$ such that $x < 18$.

Combining the Cases

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We have found that there are no solutions to the inequality when $x-18 > 0$, and the solution to the inequality when $x-18 < 0$ is all real numbers $x$ such that $x < 18$. Therefore, the solution to the inequality $\frac{x+11}{x-18} \leq 1$ is all real numbers $x$ such that $x < 18$.

Conclusion

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In this article, we have solved the inequality $\frac{x+11}{x-18} \leq 1$ algebraically and found the solution set. We have shown that the solution to the inequality is all real numbers $x$ such that $x < 18$. This is a step-by-step guide on how to solve inequalities algebraically, and we hope that it has provided a clear understanding of the process.

Final Answer

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The final answer is $\boxed{(-\infty, 18)}$.

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Introduction

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In our previous article, we solved the inequality $\frac{x+11}{x-18} \leq 1$ algebraically and found the solution set. In this article, we will answer some frequently asked questions related to solving inequalities algebraically.

Q&A

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

A: The first step in solving an inequality algebraically is to get rid of the fraction by multiplying both sides of the inequality by the denominator. However, we need to be careful when multiplying by a negative number, as it will change the direction of the inequality.

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

A: When multiplying by a negative number, we need to change the direction of the inequality. This means that if the inequality is $\leq$ or $\geq$, we need to change it to $\geq$ or $\leq$, respectively.

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

A: Solving an inequality is similar to solving an equation, but with one key difference: we are looking for a range of values that satisfy the inequality, rather than a single value that satisfies the equation.

Q: How do I know when to use interval notation to represent the solution set?

A: Interval notation is used to represent the solution set of an inequality when the solution set is a range of values. For example, if the solution set is all real numbers $x$ such that $x < 18$, we can represent it in interval notation as $(-\infty, 18)$.

Q: Can I use the same steps to solve a compound inequality?

A: Yes, you can use the same steps to solve a compound inequality. However, you need to be careful when multiplying by a negative number, as it will change the direction of the inequality.

Q: How do I know when to use a number line to visualize the solution set?

A: A number line is a useful tool to visualize the solution set of an inequality. It can help you to see the range of values that satisfy the inequality and make it easier to understand the solution set.

Common Mistakes

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Mistake 1: Not changing the direction of the inequality when multiplying by a negative number

When multiplying by a negative number, it is easy to forget to change the direction of the inequality. This can lead to incorrect solutions.

Mistake 2: Not considering all possible cases

When solving an inequality, it is essential to consider all possible cases, including when the denominator is positive or negative.

Mistake 3: Not using interval notation to represent the solution set

Interval notation is a powerful tool to represent the solution set of an inequality. Failing to use it can make it difficult to understand the solution set.

Conclusion

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Solving inequalities algebraically can be a challenging task, but with practice and patience, you can master it. Remember to get rid of the fraction by multiplying both sides of the inequality by the denominator, change the direction of the inequality when multiplying by a negative number, and use interval notation to represent the solution set. By following these steps and avoiding common mistakes, you can solve inequalities algebraically with confidence.

Final Tips

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• Always read the problem carefully and understand what is being asked.
• Use a number line to visualize the solution set.
• Check your work by plugging in values to make sure they satisfy the inequality.
• Practice, practice, practice! The more you practice, the more confident you will become in solving inequalities algebraically.