Solve The Inequality: ${ -5 \leqslant 3x + 1 \ \textless \ 19 }$

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In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities, specifically the inequality $-5 \leqslant 3x + 1 < 19$. We will break down the solution process into manageable steps, making it easier for readers to understand and apply the concepts.

What are Linear Inequalities?

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Linear inequalities are mathematical statements that compare two linear expressions. They are often represented in the form of $ax + b \leqslant c$ or $ax + b \geqslant c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear inequalities can be solved using various methods, including graphical, algebraic, and numerical approaches.

The Importance of Solving Linear Inequalities

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Solving linear inequalities is essential in various fields, including mathematics, science, engineering, and economics. In mathematics, inequalities are used to solve problems involving rates, ratios, and proportions. In science and engineering, inequalities are used to model real-world problems, such as population growth, chemical reactions, and electrical circuits. In economics, inequalities are used to analyze and predict market trends, consumer behavior, and economic growth.

Solving the Inequality $-5 \leqslant 3x + 1 < 19$

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To solve the inequality $-5 \leqslant 3x + 1 < 19$, we will follow a step-by-step approach.

Step 1: Subtract 1 from Both Sides

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The first step is to isolate the term containing the variable, $x$. We can do this by subtracting 1 from both sides of the inequality.

$$-5 \leqslant 3x + 1 < 19$$

Subtracting 1 from both sides gives us:

$$-6 \leqslant 3x < 18$$

Step 2: Divide Both Sides by 3

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Next, we need to isolate the variable, $x$. We can do this by dividing both sides of the inequality by 3.

$$-6 \leqslant 3x < 18$$

Dividing both sides by 3 gives us:

$$-2 \leqslant x < 6$$

Step 3: Write the Solution in Interval Notation

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The final step is to write the solution in interval notation. Interval notation is a way of representing a set of numbers using square brackets or parentheses.

$$[-2, 6)$$

This notation indicates that the solution set includes all numbers between -2 and 6, but does not include 6.

Conclusion

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Solving linear inequalities is an essential skill in mathematics and other fields. By following a step-by-step approach, we can solve inequalities involving linear expressions. In this article, we solved the inequality $-5 \leqslant 3x + 1 < 19$ using algebraic methods. We also discussed the importance of solving linear inequalities and provided a step-by-step guide to solving inequalities.

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 involving a linear expression, while a quadratic inequality is an inequality involving a quadratic expression.

Q: How do I solve a linear inequality with a negative coefficient?

A: To solve a linear inequality with a negative coefficient, you can multiply both sides of the inequality by -1, but be sure to reverse the direction of the inequality.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities, but be sure to check your work and verify the solution.

Additional Resources

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For more information on solving linear inequalities, check out the following resources:

• Khan Academy: Linear Inequalities
• Mathway: Linear Inequality Solver
• Wolfram Alpha: Linear Inequality Solver

By following the steps outlined in this article and using the resources provided, you can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical and real-world problems.

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In our previous article, we discussed the importance of solving linear inequalities and provided a step-by-step guide to solving the inequality $-5 \leqslant 3x + 1 < 19$. However, we know that solving inequalities can be a challenging task, and many readers have questions about the process. In this article, we will address some of the most frequently asked questions about solving linear inequalities.

Q&A: Solving Linear Inequalities

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

A: A linear inequality is an inequality involving a linear expression, while a quadratic inequality is an inequality involving a quadratic expression. For example, $2x + 3 \leqslant 5$ is a linear inequality, while $x^2 + 2x + 1 \geqslant 0$ is a quadratic inequality.

Q: How do I solve a linear inequality with a negative coefficient?

A: To solve a linear inequality with a negative coefficient, you can multiply both sides of the inequality by -1, but be sure to reverse the direction of the inequality. For example, to solve $-2x + 3 \leqslant 5$, you can multiply both sides by -1 to get $2x - 3 \geqslant -5$.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities, but be sure to check your work and verify the solution. Some calculators have built-in inequality solvers that can help you find the solution to a linear inequality.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you can use a number line and a marker to indicate the solution set. For example, to graph the inequality $x \geqslant 2$, you can mark the point 2 on the number line and shade the region to the right of 2.

Q: Can I solve a linear inequality with multiple variables?

A: Yes, you can solve a linear inequality with multiple variables. For example, to solve the inequality $2x + 3y \leqslant 5$, you can use the same steps as solving a linear inequality with one variable.

Q: How do I check my work when solving a linear inequality?

A: To check your work when solving a linear inequality, you can plug in a test value into the inequality and see if it is true. For example, to check the solution to the inequality $x \geqslant 2$, you can plug in the value 3 and see if it is true: $3 \geqslant 2$ is true.

Common Mistakes to Avoid

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When solving linear inequalities, there are several common mistakes to avoid:

• Not following the order of operations: When solving a linear inequality, be sure to follow the order of operations (PEMDAS) to ensure that you are solving the inequality correctly.
• Not checking your work: Always check your work when solving a linear inequality to ensure that the solution is correct.
• Not considering the direction of the inequality: When solving a linear inequality, be sure to consider the direction of the inequality (less than, greater than, or equal to) to ensure that the solution is correct.

Conclusion

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Solving linear inequalities can be a challenging task, but with practice and patience, you can become proficient in solving these types of inequalities. By following the steps outlined in this article and using the resources provided, you can become a master of solving linear inequalities and apply this skill to a wide range of mathematical and real-world problems.

Additional Resources

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For more information on solving linear inequalities, check out the following resources:

• Khan Academy: Linear Inequalities
• Mathway: Linear Inequality Solver
• Wolfram Alpha: Linear Inequality Solver

By following the steps outlined in this article and using the resources provided, you can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical and real-world problems.