Solve The Inequality: ${ \frac{2}{3} \leq -\frac{4}{5}(x-3) \ \textless \ 1 }$

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Introduction

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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 a specific inequality: $\frac{2}{3} \leq -\frac{4}{5}(x-3) \ \textless \ 1$. We will break down the steps involved in solving this inequality and provide a clear explanation of each step.

Understanding the Inequality

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The given inequality is $\frac{2}{3} \leq -\frac{4}{5}(x-3) \ \textless \ 1$. This inequality has two parts: $\frac{2}{3} \leq -\frac{4}{5}(x-3)$ and $-\frac{4}{5}(x-3) \ \textless \ 1$. We need to solve each part separately and then combine the solutions.

Step 1: Isolate the Variable

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To solve the inequality, we need to isolate the variable $x$. We can start by distributing the negative sign to the terms inside the parentheses: $-\frac{4}{5}(x-3) = -\frac{4}{5}x + \frac{12}{5}$.

Step 2: Simplify the Inequality

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Now that we have isolated the variable, we can simplify the inequality by combining like terms: $\frac{2}{3} \leq -\frac{4}{5}x + \frac{12}{5} \ \textless \ 1$.

Step 3: Subtract $\frac{12}{5}$ from Both Sides

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To get rid of the constant term on the right-hand side, we can subtract $\frac{12}{5}$ from both sides of the inequality: $\frac{2}{3} - \frac{12}{5} \leq -\frac{4}{5}x \ \textless \ 1 - \frac{12}{5}$.

Step 4: Simplify the Left-Hand Side

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We can simplify the left-hand side of the inequality by finding a common denominator: $\frac{10}{15} - \frac{36}{15} \leq -\frac{4}{5}x \ \textless \ \frac{5}{5} - \frac{12}{5}$.

Step 5: Simplify the Right-Hand Side

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We can simplify the right-hand side of the inequality by finding a common denominator: $-\frac{26}{15} \leq -\frac{4}{5}x \ \textless \ -\frac{7}{5}$.

Step 6: Multiply Both Sides by $-\frac{5}{4}$

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To isolate the variable $x$, we can multiply both sides of the inequality by $-\frac{5}{4}$: $\frac{65}{12} \geq x \ \textless \ \frac{35}{12}$.

Step 7: Write the Final Solution

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The final solution to the inequality is $\frac{65}{12} \geq x \ \textless \ \frac{35}{12}$.

Conclusion

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Solving inequalities involves breaking down the inequality into smaller parts, isolating the variable, and then combining the solutions. In this article, we solved the inequality $\frac{2}{3} \leq -\frac{4}{5}(x-3) \ \textless \ 1$ by following these steps. We hope that this article has provided a clear explanation of how to solve inequalities and has helped you to understand the concept better.

Frequently Asked Questions

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Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than, less than, or equal to symbols.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable, simplify the inequality, and then combine the solutions.

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 $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an inequality?

A: To graph an inequality, you need to graph the related equation and then shade the region that satisfies the inequality.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

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• [1] Khan Academy: Inequalities
• [2] Mathway: Inequalities
• [3] Wolfram Alpha: Inequalities

Note: The references and further reading section are not included in the word count.

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Introduction

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In our previous article, we discussed how to solve inequalities, including the steps involved in isolating the variable, simplifying the inequality, and combining the solutions. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A guide to help you understand and solve inequalities.

Q: What is an inequality?

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A: An inequality is a mathematical expression that compares two values, often using greater than, less than, or equal to symbols.

Q: How do I solve an inequality?

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A: To solve an inequality, you need to follow these steps:

1. Isolate the variable by adding or subtracting the same value to both sides of the inequality.
2. Simplify the inequality by combining like terms.
3. Combine the solutions by finding the intersection of the two inequalities.

Q: What is the difference between a linear inequality and a quadratic inequality?

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A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an inequality?

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A: To graph an inequality, you need to graph the related equation and then shade the region that satisfies the inequality.

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

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A: A strict inequality is an inequality that uses a strict symbol, such as $<$ or $>$, while a non-strict inequality is an inequality that uses a non-strict symbol, such as $\leq$ or $\geq$.

Q: How do I solve a system of inequalities?

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A: To solve a system of inequalities, you need to find the intersection of the two inequalities. This can be done by graphing the two inequalities and finding the region where they intersect.

Q: What is the difference between a linear programming problem and a quadratic programming problem?

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A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to a set of linear constraints. A quadratic programming problem is a problem that involves maximizing or minimizing a quadratic function subject to a set of linear constraints.

Q: How do I use a graphing calculator to solve an inequality?

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A: To use a graphing calculator to solve an inequality, you need to graph the related equation and then use the calculator's built-in functions to find the region that satisfies the inequality.

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

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

• Not isolating the variable correctly
• Not simplifying the inequality correctly
• Not combining the solutions correctly
• Not considering the direction of the inequality

Conclusion

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Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. In this article, we provided a Q&A guide to help you understand and solve inequalities. We hope that this guide has been helpful in answering your questions and providing you with the tools you need to succeed.

Frequently Asked Questions

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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 $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an inequality?

A: To graph an inequality, you need to graph the related equation and then shade the region that satisfies the inequality.

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

A: A strict inequality is an inequality that uses a strict symbol, such as $<$ or $>$, while a non-strict inequality is an inequality that uses a non-strict symbol, such as $\leq$ or $\geq$.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the intersection of the two inequalities. This can be done by graphing the two inequalities and finding the region where they intersect.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

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• [1] Khan Academy: Inequalities
• [2] Mathway: Inequalities
• [3] Wolfram Alpha: Inequalities

Note: The references and further reading section are not included in the word count.